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DIFFEEENTIAL AND INTEGBAL 
CALCULUS 



C> 



By D, A. MURRAY, Ph.D. 

Professor of Applied Mathematics in McGill University. 



INTRODUCTORY COURSE IN DIFFERENTIAL EQUA- 
TIONS, for Students in Classical and Engineer- 
ing Colleges. Pp. xvi + 236. 

A FIRST COURSE IN INFINITESIMAL CALCULUS. 
Pp. xvii + 439. 

DIFFERENTIAL AND INTEGRAL CALCULUS. Pp. 
xviii + 491. 

PLANE TRIGONOMETRY, for Colleges and Second- 
ary Schools. With a Protractor. Pp. xiii + 212. 

SPHERICAL TRIGONOMETRY, for Colleges and 
Secondary Schools. Pp. x + 114. 

PLANE AND SPHERICAL TRIGONOMETRY. In One 
Volume. With a Protractor. Pp. 349. 

PLANE AND SPHERICAL TRIGONOMETRY AND 
TABLES. In One Volume. Pp. 448. 

PLANE TRIGONOMETRY AND TABLES. In One Vol- 
ume. With a Protractor. Pp. 324. 

LOGARITHMIC AND TRIGONOMETRIC TABLES. Five- 
place and Four-place. Pp. 99. 



NEW YORK: LONGMANS, GREEN, & CO. 



DIFFERENTIAL AND INTEGRAL 
CALCULUS 



BY 



DANIEL A. MURRAY, Ph.D. 

Professor of Applied Mathematics in McGill 
University 



>**< 



LONGMANS, GREEN, AND CO< 

91 and 93 FIFTH AVENUE, NEW YORK 
LONDON, BOMBAY, AND CALCUTTA 

1908 



UBHARYof OGN-oRESSf 
I wo OoDies tte 

SEP 4 1908 

GLnSa <X_ AXc. Nu. 

_16 HI 
JOPf a? l 






Q*{n* 



Copyright, 1908, by 
LONGMANS, GKEEN, AND CO. 



All rights reserved. 



J. 8. Cushing Co. — Berwick & Smith Co. 
Norwood, Mass., U.S.A. 



PEEFACE. 

The topics in this book are arranged for primary conrses in 
calculus in which the formal division into differential calculus 
and integral calculus is deemed necessary. The book is mainly 
made up of matter from my Infinitesimal Calculus. Changes, 
however, have been made in the treatment of several topics, and 
some additional matter has been introduced, in particular that 
relating to indeterminate forms, solid geometry, and motion. The 
articles on motion have been written in the belief that familiarity 
with the notions of velocity and acceleration, as treated by the 
calculus, is a great advantage to students who have to take, 
mechanics. 

Part of the preface of my Infinitesimal Calculus applies equally 
well to this book. Its purpose is to provide an introductory 
course for those who are entering upon the study of calculus 
either to prepare themselves for elementary work in applied 
science or to gratify and develop their interest in mathematics. 
Little more has been discussed than what may be regarded as the 
essentials of a primary course. An attempt is made to describe 
and emphasise the fundamental principles of the subject in such 
a way that, as much as may reasonably be expected, they may 
be clearly understood, firmly grasped, and intelligently applied 
by young students. There has also been kept in view the devel- 
opment in them of the ability to read mathematics and to prose- 
cute its study by themselves. 

\Yith regard to simplicity and clearness in the exposition of 
the subject, it may be said that the aim has been to write a book 
that will be found helpful by those who begin the study of 
calculus without the guidance and aid of a teacher. For these 
students more especially, throughout the work suggestions and 
remarks are made concerning the order in which the various 



VI PREFACE. 

topics may be studied, the relative importance of the various 
topics in a first study of calculus, the articles that must be 
thoroughly mastered, and the articles that may advantageously 
be omitted or lightly passed over at its first reading, and so on. 

The notion of anti-differentiation is presented simultaneously 
with the notion of differentiation, and exercises thereon appear 
early in the text ; but when integration is formally taken up the 
idea of integration as a process of summation is considered before 
the idea of integration as a process which is the inverse of 
differentiation. There is considerable difference of opinion as to 
the propriety or the advantage of this order. The decision to 
follow it here has been made mainly for the reason that students 
appear — at least so it seems to me, but other teachers may have 
a different experience — to understand more clearly and vividly 
the relation of integration to many practical problems when the 
summation idea is put in the forefront. In teaching, the one 
order can be taken as readily as the other. 

In several technical schools the time assigned to calculus is 
not sufficient for a fair study of Taylor's theorem. What may 
be regarded as the irreducible requisite for a slight working 
acquaintance with Taylor's and Maclaurin's series is indicated 
at the beginning of Chapter XV., and may be taken at an early 
stage in the course. 

An explanation of hyperbolic functions can be made more 
naturally and more fully, perhaps, in a course in calculus than 
in any other course in elementary mathematics. ' Eor this reason, 
and also because students will meet them in their later work and 
reading, a note on these functions appears in the latter part of 
the book. 

Owing to the pressure of other subjects the time allotted to 
mathematics in quite a number of technical schools is rather 
brief. Where this is the case, and where there is a lack of 
maturity in the students, it is better not to try to cover too 
much ground, but to lay stress on fundamental principles, to' 
drill in the elementary processes, and to train in making simple 
applications. Thus this book, small as it may be regarded even 
for a short course, contains more matter than can be thoroughly 
studied in the few months allotted to calculus in colleges and 



PBEFACE. vii 

technical schools where such conditions exist. Several topics, 
however (for example, the investigation of series), which in some 
cases are not studied by technical students owing to lack of time, 
are very important, particularly for those who take a first course 
in the calculus as an introduction to a more extended study of 
the subject aud as part of the preparation necessary for more 
advanced work in mathematics. For the sake of these students 
more especially, but not exclusively on their account, many definite 
references for collateral reading or inspection are given throughout 
the text. 

It is hoped that these references will add to the helpfulness 
of the book. With but very few exceptions those are chosen 
which are easily accessible to all college students. Some of the 
references will aid the learner by presenting an idea of the text 
in the words of another ; but the larger number of them are 
intended to direct students to places where they will either re- 
ceive fuller information or be impressed with some of the impor- 
tant modern ideas of mathematics. Turning up such references 
as these will increase the mathematical interest of the student 
and widen his outlook. It will also help to train the pupils in 
the use of mathematical literature, and, by arousing and exercis- 
ing their critical faculties, will greatly benefit those who may 
intend to teach mathematics in the secondary schools. Of course 
the lists of references are not exhaustive, and, while care has 
been taken in making them, it is to be expected that several 
other equally serviceable lists can be arranged. It is intended 
that these lists shall be revised and supplemented by those who 
may use the book. 

Not many examples involving a technical knowledge of engi- 
neering, physics, or chemistry have been inserted. Few young 
students understand examples of this kind without considerable 
explanation, and thus it seems better to refer the pupils to the 
more specialised text-books dealing with calculus (for instance, 
those of Perry, Young and Linebarger, and Mellor), which contain 
many examples of a technical character. 

For learners who can afford but a minimum of time for this 
study the essential articles of a short course are indicated after 
the table of contents. 



Vlll PREFACE. 

I take this opportunity of thanking Mr. T. Ridler Davies, 
Lecturer in Mathematics at McG-ill University, for his kind 
assistance in the revision of the proof sheets. 

D. A. MURRAY. 
July 6, 1908. 



CONTENTS. 

DIFFERENTIAL CALCULUS. 

CHAPTER I. 
Introductory Problems. 



2. Speed of a moving tram 2 

3. To determine the speed of a falling body 2 

4. To determine the slope of a tangent 5 

5. To determine the area of a plane figure 10 

6. To find a function when its rate of change is known . . .11 
To find the equation of a curve when its slope is known . . .11 

7. Elementary notions used in infinitesimal calculus . . . .11 

CHAPTER II. 

Algebraic Notions which are frequently used in the Calculus. 

8. Variables 13 

9. Functions 14 

10. Constants . 16 

11. Classification of functions ........ 16 

12. Notation 18 

13. Graphical representation of functions of one variable ... 19 

14. Limits 20 

15. Notation 23 

15 a. Continuous variation. Interval of variation .... 24 

16. Continuous functions. Discontinuous functions .... 25 

CHAPTER III. 

Infinitesimals, Derivatives, Differentials, Anti-derivatives, and 
Anti-differentials. 

18. Infinitesimals, infinite numbers, finite numbers .... 28 

19. Orders of magnitude. Orders of infinitesimals. Orders of infinites 29 

20. Changes in the variable and the function 30 

21. Comparison of these corresponding changes ..... 31 

ix 



X CONTENTS. 

ART. PAGE 

22. The derivative of a function of one variable 32 

23. Notation 35 

24. The geometrical meaning and representation of the derivative of a 

function 37 

25. The physical meaning of the derivative of a function ... 39 

26. General meaning of the derivative : the derivative is a rate . . 40 

27. Differentials 42 

21 a. Anti-derivatives and anti-differentials 45 

CHAPTER IV. 
Differentiation of the Ordinary Functions. 

General Besults in Differentiation. 

29. The derivative of the sum of a function and a constant, say 

<p(x) + c 46 

30. The derivative of the product of a constant and a function, say 

ccp(x) 48 

31. The derivative of the sum of a finite number of functions . . 49 

32. The derivative of the product of two or more functions . . .50 

33. The derivative of the quotient of two functions .... 52 

34. The derivative of a function of a function 54 

35. The derivative of one variable with respect to another when both 

are functions of a third variable 55 

36. Differentiation of inverse functions 56 

Differentiation of Particular Functions. 
A. Algebraic Functions. 

37. Differentiation of u n . . . 56 

B. Logarithmic and Exponential Functions. 

38. Note. To find lim,^ (l + — X* 61 

39-41. Differentiation of log a w, a« u v 62-66 

C. Trigonometric Functions. 
42-48. Differentiation of sin u, cos w, tan u, cot w, sec w, esc u, vers u 66-71 

D. Inverse Trigonometric Functions. 

49-55. Differentiation of sin _1 M, cos _1 m, tan -1 u, c^t -1 n, see -1 u, 

esc -1 u, vers -1 u 71-75 

56. Differentiation of implicit functions : two variables ... 75 



CONTENTS. XI 



CHAPTER V. 

S031E Geometrical, Physical, and Analytical Applications. 
Geometric Derivatives and Differentials. 

ABT. PAGE 

59. Slope of a curve at any point : rectangular coordinates . . .79 

60. Angles at which two curves intersect 81 

61. Equations of the tangent and normal drawn at a point on a curve . 88 

62. Lengths of tangent, subtangent, normal, and subnormal : rectangu- 

lar coordinates .......... 84 

63. Slope of a curve at any point : polar coordinates .... 87 

64. Lengths of tangent, subtangent, normal, and subnormal : polar 

coordinates 88 

65. Applications involving rates 90 

66. Small errors and corrections ; relative error ..... 92 
66 a. Applications to algebra . . .93 

67. Geometric derivatives and differentials . . . . . 95-102 



CHAPTER VI. 
Successive Differentiation. 

68. Successive derivatives 103 

69. The wth derivative of some particular functions .... 108 

70. Successive differentials 109 

71. Successive derivatives of y with respect to x when both are func- 

tions of a third variable 109 

72. Leibnitz's theorem 110 

73. Application of differentiation to elimination Ill 

CHAPTER VII. 
Further Analytical and Geometrical Applications. 

74. Increasing and decreasing functions 113 

75. Maximum and minimum values of a function. Critical points on 

the graph, and critical values of the variable . . . .114 

76. Inspection of the critical values of the variable for maximum or 

minimum values of the function . . . . . . .117 

77. Practical problems in maxima and minima 121 

78. Points of inflexion : rectangular coordinates 125 

CHAPTER VIII. 
Differentiation of Functions of Several Variables. 

79. Partial derivatives. Notation 128 

80. Successive partial derivatives ........ 131 



Xll CONTENTS. 

ART. PAGE 

81. Total rate of variation of a function of two or more variables . 132 

82. Total differential 134 

83. Approximate value of small errors 136 

84. Differentiation of implicit functions ; two variables . . . 137 

85. Condition that an expression of the form Pdx + Qdy be a total 

differential 138 

86. Illustrations: partial differentials, total differentials, partial de- 

rivatives. Illustration A. ....... . 139 

87. Illustration B 140 

88. Illustration C 141 

CHAPTER IX. 
Change or Variable. 

89. Change of variable 143 

90. Interchange of the dependent and independent variables . . 143 

91. Change of the dependent variable 144 

92. Change of the independent variable 145 

93. Dependent and independent variables both expressed in terms of a 

single variable 146 

CHAPTER X. 

Concavity and Convexity. Contact and Curvature. Evolutes 
and' Involutes. 

94. Concavity and convexity : rectangular coordinates . . . 148 

95. Order of contact 149 

96. Osculating circle 152 

97. The notion of curvature 153 

98. Total curvature. Average curvature. Curvature at a point . 154 

99. The curvature of a circle 155 

100. To find the curvature at any point of a curve : rectangular coordi- 

nates ............ 155 

101. The circle of curvature at any point of a curve .... 156 

102. The radius of curvature : polar coordinates 159 

103. Evolute of a curve 160 

104. Properties of the evolute 161 

105. Involutes of a curve 164 

CHAPTER XL 

Rolle's Theorem. Theorems of Mean Value. Approximate 
Solution of Equations. 

107. Rolle's theorem 166 

108. Theorem of mean value 169 



CONTENTS. 



Xlli 



ART. 

109. 
110. 
111. 
112. 
113. 



Approximate solution of equations . . • 

Theorem of mean value derived from Rolle's theorem 
Another form of the theorem of mean value . 
Second theorem of mean value .... 
Extended theorem of mean value .... 



PAGE 

171 
174 
175 
176 
177 



ir 



CHAPTER XII. 
Indeterminate Forms. 



114. Indeterminate forms 

115. Classification of indeterminate forms 

116. Generalized theorem of mean value 



Evaluation of functions when they take the form - 





118. Evaluation of functions when they take the form ~ 

119. Evaluation of other indeterminate forms 



180 
181 
182 

183 

185 

187 



CHAPTER XIII. 
Special Topics relating to Curves. 

120. Eamily of curves. Envelope of a family of curves . . . 190 

121. Locus of ultimate intersections of the curves of a family . . 191 

122. Theorem 193 

123. To find the envelope of a family of curves having one parameter . 194 

124. Envelope of a family of curves having two parameters . . . 197 

125. Rectilinear asymptotes 199 

126. Asymptotes parallel to the axes . . . . . . . 201 

127. Oblique asymptotes 203 

128. Rectilinear asymptotes : polar coordinates 205 

129. Singular points 206 

130. Multiple points 206 

131. To find multiple points, cusps, and isolated points . . . 209 

132. Curve tracing 211 

133. Note supplementary to Art. 127 212 



CHAPTER XIV. 
Applications to Motion. 



134. Speed, displacement, velocity .... 

135. To find velocity of a point moving on a curve 

136. Composition of displacements .... 

137. Resolution of a displacement into components 

138. Composition and resolution of velocities 

139. Component velocities of a point moving on a curve 



214 
216 
216 

218 
219 
220 



XIV 



CONTENTS. 



ART. PAGE 

140. Acceleration .223 

141. Acceleration : particular cases . . . . . . . 224 



CHAPTER XV. 



Infinite Series. 



142. Infinite series : definitions, notation 

143. Questions concerning infinite series 

144. Study of infinite series 

145. Definitions. Algebraic properties of infinite series 

146. Tests for convergence ...... 

147. Differentiation of infinite series term by term 

148. Examples in the differentiation of series 



230 
231 
233 
234 
237 
240 
240 



CHAPTER XVI. 
Taylor's Theorem. 

150. Derivation of Taylor's theorem 242 

151. Another form of Taylor's theorem 246 

152. Maclaurin's theorem and series 247 

153. Relations between the circular functions and exponential functions 250 

154. Another method of deriving Taylor's and Maclaurin's series . 252 

155. Application of Taylor's theorem to the determination of condi- 

tions for maxima and minima 254 

156. Application of Taylor's theorem to the deduction of a theorem on 

the contact of curves 255 

157. Applications of Taylor's theorem in elementary algebra . . 256 



CHAPTER XVII. 
Applications to Surfaces and Twisted Curves. 



158. Introductory ....... 

159. Tangent line to a twisted curve 

160. Equation of plane normal to a skew curve 

161. Tangent lines and tangent plane to a surface 

162. Normal line to a surface .... 

163. Equations of tangent line and normal plane to a skew curve 



257 
259 
260 
262 
264 
266 



CONTENTS. XV 



INTEGRAL CALCULUS. 

CHAPTER XVIII. 
Integration. 

ART. PAGE 

164. Integration and integral defined. Notation . . . ... 269 

165. Examples of the summation of infinitesimals . . . .271 

166. Integration as summation. The definite integral .... 215 

167. Integration as the inverse of differentiation. The indefinite integral 
Constant of integration. Particular integrals' .... 281 

168. Geometric or graphical representation of definite integrals 
Properties of definite integrals 284 

169. Geometric or graphical representation of indefinite integrals 
Geometric meaning of the constant of integration .... 287 

170. Integral curves 289 

171. Summary 290 

CHAPTER XIX. 

Elementary Integrals. 

173. Elementary integrals 293 

174. General theorems in integration 294 

175. Integration aided by substitution . 296 

176. Integration by parts 298 

177. Further elementary integrals 301 

178. Integration of f(x) dx when f(x) is a rational fraction . . . 305 

179. Integration of a total differential 309 

CHAPTER XX. 

Simple Geometrical Applications of Integration. 

181. Areas of curves : Cartesian coordinates 313 

182. Volumes of solids of revolution 320 

183. Derivation of the equations of curves 324 

CHAPTER XXI. 

Integration of Irrational and Trigonometric Functions. 

Integration of Irrational Functions. 

185. The reciprocal substitution 327 

186. Differential expressions involving y/a + bx 328 



XVI CONTENTS. 



187. A. Expressions of form F(x, Vx 2 + ax -f b)dx. B. Expressions 

of form F(x, V-x 2 + ax + b)dx 329 

188. To find (x m (a + bx n ydx 332 

Integration of Trigonometric Functions. 

189. Algebraic transformations 336 

190. Integrals reducible to \ F(u)du, in which u is one of the trigo- 

nometric ratios .......... 337 

191. Integration aided by multiple angles 338 

192. Reduction formulas 339 

CHAPTER XXII. 
Approximate Integration. Mechanical Integration. 

193. Approximate integration of definite integrals .... 344 

194. Trapezoidal rule for measuring areas and evaluating definite inte- 

grals 344 

195. Parabolic rule for measuring areas and evaluating definite integrals 346 

196. Mechanical devices for integration 348 

CHAPTER XXIII. 

Integration of Infinite Series. 

197. Integration of infinite series term by term 350 

198. Expansions obtained by integration of known series . . . 350 

199. Approximate integration by means of series 353 

CHAPTER XXIV. 
Successive Integration. Multiple Integrals. Applications. 

201. Successive integration : one variable. Applications . . . 355 

202. Successive integration : several variables . 357 

203. Finding areas : rectangular coordinates ....«„ 359 

204. Finding volumes : rectangular coordinates . 360 

205. Finding volumes : polar coordinates ....„* 363 

CHAPTER XXV. 

Further Geometrical Applications of Integration. 

207. Volumes of solids of known cross-section . . . . . 365 

208. Areas : polar coordinates 367 

209. Lengths of curves : rectangular coordinates . . ..... • 370 



CONTENTS. xvii 

AKT. PAGE 

210. Lengths of curves : polar coordinates 373 

211. Areas of surfaces of revolution 374 

212. Areas of surfaces z =f(x, y) 378 

213. Mean values 380 

214. Note to Art. 104 384 

CHAPTER XXVI. 

Note ox Centre or Mass and Moment of Inertia. 

215. Mass, density, centre of mass ....... 385 

210. Moment of inertia. Radius of gyration 390 

CHAPTER XXVII. 

Differential Equations. 

217. Definitions. Classifications. Solutions 394 

218. Constants of integration. General solutions. Particular solutions 395 

Equations of the First Order. 

219. Equations of the form f(x)dx + F(y)dy = 395 

220. Homogeneous equations . . . . • . . . . 396 

221. Exact differential equations. Integrating factors .... 396 

222. The linear equation 397 

223. Equations not of the first degree in the derivative : 

The form x =f(y, p) ; the form y = f(x,p) ; Clairaut's equation 398 

224. Singular solutions 400 

225. Orthogonal trajectories ......... 401 

Equations of the Second and Higher Orders. 

226. Linear equations with constant coefficients. Homogeneous linear 

equations ........... 406 

227. Special equations of the second order : 

g=/« ; /(g,I,^0 ;/ (g,|„) = . . . 409 



APPENDIX. 

Note A. Hyperbolic functions 413 

Note B. Intrinsic equations ........ 423 

Note C. Length of a curve in space 427 

Collection of Examples 431-450 



XVlll CONTENTS. 

PAGE 

Integrals for Eeview Exercises and for Reference . . 451-458 

Figures 459-464 

Answers 465 

Index . . . . 485 



SHORT COURSE 

FOR STUDENTS HAVING A MINIMUM OF TIME 
(The Roman numerals refer to chapters, the Arabic to articles.) 

II. ; III. ; IV. ; V. 57-65 ; VI. 68-70 ; VII. ; VIII. 79-84, 86 ; IX. (if 
time permits) ; X. ; XI. 108, 109 ; XIII. (part) ; XV. ; XVI. 149, 
154, Exs. 150-152 ; XVIII. ; XIX. ; XX. ; XXI. 184-186, (188-192, 
if time permits) ; XXII. ; XXIII. ; XXIV. ; XXV. 206-211, 213. 
Recommended for students looking forward to engineering and to 
courses in mechanics : XIV., XXVI. 



DIFFERENTIAL CALCULUS 
CHAPTER I. 

INTRODUCTORY PROBLEMS. 

1. The infinitesimal calculus is one of the most powerful mathe- 
matical instruments ever invented.* Many practical problems can 
be solved by its means with wonderful ease and rapidity. Even 
a slight acquaintance with the calculus is very helpful in the study 
of many other subjects, for example, geometry, astronomy, physics, 
and engineering; and the fullest knowledge possible about the 
calculus is necessary for advance in these subjects. Some of 
the higher branches of mathematics consist largely of special 
investigations in the infinitesimal calculus and extensions of its 
principles, methods, and applications.! 

In this book the fundamental notions and principles of the 
calculus are, to a certain extent, explained, and applications are 
made to the solution of some simple practical problems. As a 
preliminary to the study there is in this chapter a discussion of 
a few problems. This discussion introduces in an informal way 
the notions and principles and methods which are at the founda- 
tion of the infinitesimal calculus, and also provides material which 
serves to illustrate a few of the articles that follow.! 

* The calculus is divided into two parts, the differential calculus and the 
integral calculus. Concerning its invention see Art. 164, note. 

t The word "infinitesimal" serves to distinguish the subject from other 
branches of mathematics, such as the calculus of finite differences, the cal- 
culus of variations, the calculus of quaternions, etc. 

1 An important fact in the history of the calculus is that the problems in 
Arts. 3-6 were the occasion of the invention and development of some divi- 
sions of the subject. 

1 



2 DIFFERENTIAL CALCULUS. [Ch. I. 

Note. A knowledge of the meaning of the term speed or rate of motion is 
presupposed in the following two articles. If a body moves through equal 
distances in equal times, it is said to have uniform speed. The average speed 
of a body during the time that it is moving through a certain distance, is the 
uniform speed at which a body will pass over that distance in that time. 
For instance, if a bicyclist wheels 36 miles in 3 hours, his average speed is 
12 miles per hour ; if a body moves through 45 feet in 5 minutes, its average 
speed is 9 feet per minute. The number which indicates the average speed 
of a body while it is moving through a certain distance, is the ratio of the 
number of units of length in the distance to the number of units of time spent 
during the motion. In other words, the measure of the speed is the ratio of 
the measure of the distance to the measure of the corresponding time. Thus, 
in the instances above, 12 = 36 : 3, 9 = 45 : 5. 

Any reader of this book knows what is meant by the statements that a 
train is running at a particular instant at the rate of 30 miles an hour, and 
that at another instant, some minutes later say, it is running at the rate of 40 
miles an hour. This notion, viz. the speed of a moving body at a par- 
ticular instant, will be developed further by the examples that follow. 

2. Speed of a moving train. Suppose that a person is standing 
by a railway and wishes to ascertain the speed at which a train 
is going by him. A way to determine this speed approximately 
would be to find the distance passed over in five seconds by the 
train, or by a definite mark on the train, say a vertical line. (The 
place where the observer stands may be at one end of, or upon, 
the measured distance.) If the observer knew the distance passed 
over in three seconds, he would get the speed more accurately ; 
yet more accurately, if lie knew the distance passed over in one 
second; more accurately still, if he knew the distance passed 
over in half a second; and so on. The point to be noted and 
emphasised in this illustration is this : the less the time and the 
corresponding distance that can be observed, the more nearly will 
the observer obtain the actual speed of the train just at the 
moment when it is passing him. 

3. To determine the speed of a falling body. Let a body fall 
vertically from rest. It is known that in t seconds from the 
time of starting, the body passes through \gf feet. (Here g 
denotes a number whose approximate value is 32.2.) That is, if 
s denotes the number of feet through which the body falls in t 
seconds, s = ±gt\ 



2,3.] 



INTBOBUCTOBY PBOBLEMS. 



As the body descends its speed is continually changing and grow- 
ing greater; but at any particular instant it has some definite 
speed. Let it be required to find the speed after it has been 
falling for : (a) 4 seconds ; (6) t 1 seconds. 

(a) To find speed after the body has been falling from rest for 4 seconds. 
A method of getting an approximate value of this speed is as follows, Find 
the distance through which the body would fall in 4 seconds ; then find the 
distance through which it would fall in a little more than 4 seconds. There- 
from deduce the average value of the speed from the end of the fourth second 
to the last instant (Note, Art. 1). This average speed may be taken as an 
approximate value of the speed at the end of the fourth second. The smaller 
the interval of time which is taken after the fourth second, the more nearly 
will the average speed for the interval be equal to the actual speed just at 
the end of the fourth second. This is also apparent from the following 
calculations : 



a 




® rA 


Corresponding 






°^ 


Length of fall, 


3§a 


increase m 


Average s 


peed during increased 


g § 


in feet. 


g^ § 


distance, 


time, 


in feet per second. 


33 "0 




§ 31 M 
g -2 c 


in feet. 






3 
P 




j^s 








4. 


8? 














4.1 


8.405 g 


.1 


.405 g 


4.05 


or 130.41 


4.01 


8.04005 


.01 


.040050 


4.005 


128.961 


4.001 


8.0040005^ 


.001 


.0040005 


4.0005 


128.8161 


4.0001 


8.000400005 g 


.0001 


.0004000050 


4.00005 


128.80161 


l + h 


(8+4*+i**)0 


h 


(4A+iA*)0 


(* + |> 


128.8 + 16.1x7* 



It is evident that the less the increase given to the 4 seconds, the more 
nearly does the average speed during this additional time approach to 128.8 
feet per second. The last line of the table shows that, no matter how short 
a time h may be, the average speed during this time has a definite value, 
namely (128.8 + 16.1 x h) feet per second. The number in brackets becomes 
more and more nearly equal to 128.8 when h is made smaller and smaller ; the 
difference between it and 128.8 can be made as small as one pleases, merely 
by decreasing h, and will become still less when h is further diminished. 
Since the number (128.8 + 16.1 x h) behaves in this way, the speed of the 
falling body at the end of the fourth second is manifestly 128.8 feet per 
second. 



4 DIFFERENTIAL CALCULUS, [Ch. I. 

(6) To find the speed after the body has been falling for h seconds. Let 

si denote the distance in feet through which the body has fallen in the t\ 

seconds. It is known that _ _ i nt 2 n\ 

si = i gh • (L) 

Let Ah (read " delta £i") denote any increment given to h, and Asi denote 
the corresponding increment of Sj.. 

Note 1. Here Ah does not mean A x t\. The symbol A is used with a 
quantity to denote any difference, change, or increment, positive or negative 
(i.e. any increase or decrease), in the quantity. Thus Ax and Ay denote 
" increment of as," " increment of ?/," " difference in x," " difference in y." 

Then si + Asi = ±g(t 1 + Ah) 2 . (2) 

Hence, by (1) and (2), Asi = gh ■ Ah + \ g(&h) 2 - 

■•■ ^i-^ + ^-A^. (3) 

Ah 

Here — ^ is the average speed for the time Ah and the corresponding 

A h Asi 

distance Asi. Now the smaller Ah is taken, the more nearly will ~rr 

approximate to the actual speed which the falling body has at the end of 
the t\th second. But when Ah is taken smaller and smaller (in other words, 
when Ah approaches nearer and nearer to zero), the second member of equa- 
tion (3) approaches nearer and nearer to gh. Equation (3) also shows that 

— — can be made to differ as little as one pleases from gh, merely by taking 

Ah 

Ah small enough. Hence it is reasonable to conclude that at the end of the 

tfith second 

the speed of the falling body = gh feet per second. (4) 

Here h may be any value of t. So it is usual to express conclusion (4) 
thus : the speed of a body that has been falling for t seconds is gt feet per 
second. This result (speed = gt feet per second) is a general one, and can 
be applied to special cases. Thus at the end of the fourth second the speed 
is g x 4 or 128.8 feet per second, as found in (a) ; at the end of 10 seconds 
the speed is 10 g or 322 feet per second. 

The two principal points to be noted in this illustration are : 

(1) No matter what the value of At x may be, or how small A£ x 

As 
may be, the quantity — - 1 has a definite value, namely, gt x + \ g • A£ 3 ; 

(2) When A^ is taken smaller and smaller, — - gets nearer and 

nearer to gt^; and the difference between them can be made as 
small as one pleases by giving A^ a definite small value; this 
difference remains less than the assigned value when A^ further 
decreases. 



4.] 



INTRODUCTORY PROBLEMS. 



Note 2. The definite small value referred to in (2) can be easily found. 
For example, suppose that -^ is to differ from gt x by not more than k say (k 
being any small quantity, as a millionth, or a million-millionth). 



Then ^ - gh < fc 



But 



^-^i = i^;A«i by (3). 

2& 



. \ J- g • At 1< k; accordingly A«i < 

Note 3. It should be observed, as shown by equation (3), that the value of 

— ^ depends upon the values of both t\ and A^. On the other hand, the 

At i Asi 

value to which — tends to become equal as Ati decreases, depends (see (4)) 

upon t\ alone. The quantity A*i is any increment whatever of fr, but it does 
not depend upon the value of t\. 

4. To determine the slope of the tangent to the parabola y = x 2 : 
(a) at the point whose abscissa is 2 ; (b) at the point whose abscissa 
is x v 

(a) Let VOQ, Fig. 1, be the 
parabola y = x 2 , and P be the 
point whose abscissa is 2. 
Draw the secant PQ. If PQ 
turns about P until Q coin- 
cides with P, then PQ will 
take the position PT and be- 
come the tangent at P. The 
angle QPR will then become the angle PPT. 

Note 1. This conception of a tangent to a curve has probably been 
already employed by the student in finding the equations of tangents to circles, 
parabolas, ellipses, and hyperbolas. The process generally followed in the 
analytic treatment of the conic sections is as follows : The equation of the 
secant PQ is found subject to the condition that P and Q are on the curve ; 
then Q is supposed to move along the curve until it reaches P. The resulting 
form of the equation of the secant is the equation of the tangent at P. The 
calculus method (now to be shown) of finding tangents to curves is preferred 
by some teachers of analytic geometry ; e.g. see A. L. Candy, Analytic 
Geometry, Chap. V. 

Draw the ordinates LP and MQ ; draw PR parallel to OX. 
Let PR be denoted by Ax, and RQ by Ay. Then the slope of 

the secant PQ is ^ f For tan RPQ = ^2.^ 
^ Ax \ PR J 




Fig. 1. 



6 



DIFFERENTIAL CALCULUS. 



[Ch. I. 



The following table shows the value of — for various values 

of Ax. 





Corresponding 


Ax 


Ay 


Corresponding 

value of ^- 

Ax 




value of y. 


(Increase over®). 


(Increase over y). 


2. 


4. 








2.1 


4.41 


.1 


.41 


4.1 


2.01 


4.0401 


.01 


.0401 


4.01 


2.001 


4.004001 


.001 


.004001 


4.001 


2.0001 


4.00040001 


.0001 


.00040001 


4.0001 


2 + h 


4 + 4 h + h* 


h 


4 h + h 2 


4 + ft 



It is apparent from this table that the less Ax is, the more nearly does 
— y~ approach the value 4. The last line shows that, no matter how small Ax 

Ax Av 

(or h) may be, — 2 has a definite value, namely 4 + h. This number becomes 

Ax 
more and more nearly equal to 4 when h is made less and less ; the difference 
between it and 4 can be made as small as one pleases, merely by decreasing h 
to a certain definite value, and will continue to be as small or smaller when 
h is further diminished. Because the number 4 + h behaves in this way, 

it is evident that — ^ will reach the value 4 when Ax decreases to zero. 

Ax 
Accordingly the slope of the tangent FT is 4 ; and hence angle TFB or 
PWL is 75° 57' 49". 

(b) To determine the slope of the tangent at the point whose 
abscissa is x v 

Let (Fig. 1) P be the point (xi, y{). Draw the secant PQ, and the 
ordinates PL and QM ; draw PR parallel to OX. Let PR, the difference 
between the abscissas of P and Q, be denoted by Axj., and let RQ, the 
difference between the ordinates of P and Q, be denoted by Ayi. Then 



tangent QPR 



RQ 

PR 



AJ/l. 

Axi 



If Q be moved along the curve toward P, the secant PQ will approach 
the position of P7 7 , the tangent at P ; at last, when Q reaches P, the secant 
PQ becomes the tangent PT. As Q approaches P, Axi becomes less and 
less, and when Q reaches P, Axi becomes zero. Conversely, as Axi decreases, 
PQ approaches the position PT. Accordingly, the slope of the tangent PT 
can be determined by finding what the slope of the secant PQ, namely J^, 
approaches when Axi approaches zero. 



Axi 



* 

4.] INTRODUCTORY PROBLEMS, 7 

Hence, on subtraction, Ayi = 2x 1 > Ax\ + (Axi) 2 . (1) 

m . m *yi = 2x 1 + Axi. (2) 

Axi 

This equation shows that -^ approaches nearer to 2 Xi when Axi decreases. 

Av AjCl 

It also shows that -^± can be made to differ as little as one pleases from 2 Xi, 

Axi 
merely by taking Axi small enough, and that this difference will become 
smaller when Axi is further diminished. (For instance, if it is desired that 

— y± — 2 Xi be less than any positive small quantity, say e, it is only necessary 

Axi 

to take Axi less than e.) Accordingly, 

the slope of PT (the tangent at P) = 2 xi. (3) 

The two principal points to be noted in this illustration are : 

(1) No matter what the value of Ax 1 may be, or how small Aa^ 

may be, the quantity — — has a definite value, namely 2x x -\- Ax v 

1 Aw 

(2) When A.x x decreases, the quantity — — approaches the 

A?/ x 

value 2x-,\ the difference between -^ and 2x x can be made as 

A#! 

small as any number that may be assigned, by giving Ax 1 a 

definite small value ; this difference remains less than the 

assigned value when Ax ± further decreases. 

Ay-t 

Note 1. The value of — — , as shown by Equation (2), depends upon the 

A 

values of both xi and Axi. On the other hand, the value to which — — 

Axi 
tends to become equal as Axi decreases, depends (Equation (3)) upon Xi 
alone. The value of Axi does not depend upon the value of x\ ; for Q 
(Fig. 1) may be taken anywhere on the curve. 

Xote 2. The method used in getting result (3) does not depend upon 
the particular value of x\. The result is perfectly general, and may be 
expressed thus : " the slope of the curve y = x 2 is 2 x." This general result 
can be used for finding the slope at particular points on the curve. For 
instance, if X\ = 2, the slope is 4, as found in («) ; if X\ =— 1, the slope 
is — 2, and accordingly, the angle made by the tangent with the x-axis is 
116° 34'. (It is advisable to make a figure showing this.) 

Note 3. In the infinitesimal calculus, as well as in other branches of 
mathematics, it is very important for the student always to have a clear 



8 DIFFERENTIAL CALCULUS. [Ch. I. 

understanding of the meaning of the operations which he performs with 
numbers, and to interpret rightly the numerical results obtained by these oper- 
ations. Thus, if it is stated that 6 men work 5 days at 2 dollars per day each, 
the numbers 6, 5, and 2 are treated by the operation called multiplication, 
and the number 60 is obtained. The calculator then applies, or interprets, 
this numerical result as meaning, not 60 men, or 60 days, buc that the men 
have earned 60 dollars. In the curve above, y = x 2 . This does not mean 
that at any point on the curve the ordinate is equal to the square on the 
abscissa, i.e. a length is equal to an area. By y = x 2 it is meant that the 
number of units of length in any ordinate is equal to the square of the num- 
ber of units of length in the corresponding abscissa. Again, the result in 
Equation (3) does not mean that the slope of FT is twice OL. The result 
means that the number which is the value of the trigonometric tangent of 
the angle TPB is twice the number of units of length in OL. 

Many persons who can perform operations of the calculus easily and 
accurately, cannot correctly or confidently interpret the results of these 
operations in concrete practical problems in geometry, physics, and engi- 
neering. Thus, some engineers who have had a fairly extended course in 
calculus discard it when possible, and solve practical problems by much 
longer and more laborious methods. Such a misfortune will not happen to 
those who early get into the habit of giving careful thought to finding out the 
real meaning of the operations and results of the calculus. They will not 
only "understand the theory," but they can use the calculus as a tool with 
ease and skill. 

Note 4. In Fig. 1 let a point Qi be taken on the curve to the left of P, 
and draw the secant Q\P. (The drawing for this note is left to the student.) 
It is obvious from the figure that the same tangent FT is obtained, whether 
the secant Q±P revolves until Q^ reaches P, or QP revolves until Q reaches 
P. This may also be deduced algebraically. Let the coordinates of Qi be 
Xi — Aasi, y\ — A?/i. [Here the A^i and A?/i are not necessarily the same in 
amount as the Axi and A?/i in (&).] Draw the ordinate QiMi. Then 

y 1 (=LP)=x 1 2 , 

Vi - Ayi (= Jfi#i) = Qd - Axi)2. 

Whence, it follows that — — = 2 x\ — A.X\. 

Accordingly, when Axi approaches zero, — — approaches the value 2 X\. 

Note 5. Thoughtful beginners in calculus are frequently, and not un- 
naturally, troubled by the consideration that when A#i (Art. 3 b) is diminished 

to zero, ~ has. the form -; and likewise, when Axi (Art. 4 6) becomes 
A£i 

zero, -^ becomes -. It is true that K is indeterminate in form ; and, if 
Axi ' ' 



4-] UVTBODUCTOBY PROBLEMS. 9 

it is presented icithout any information being given concerning the whence 
and the wherefore of its appearance, a value for it cannot be determined. 
In the cases in Arts. 3, 4, however, there is given information which makes 

it possible to tell the meaning of the quantity - that appears at the final stage 

of each of these problems. In these cases one knows how the quantities 

Asi A?/i 

AT Ax ar€ behavin 9 wben A *i and Ax i respectively are approaching 

zero ; and by means of this knowledge he can confidently and accurately 
state what these ratios will become when Ah and A:*^ actually reach zero.* 

Note 6. Moreover, it should be carefully noted that at the final stages 

in the solution of the problems in Arts. 8 and 4, — - is not regarded as a 

fraction composed of two quantities, Asi and Ati, but as a single quantity, 

namely the speed after t\ seconds ; likewise, that — — is then not regarded 

as a fraction at all, but as a single quantity, namely the slope of the tangent 
at P. 

Note 7. The student should not be satisfied until he clearly perceives, 
and understands, that the method employed in solving the problems in 
Arts. 3 and 4 is not a tentative one, but is general and sure, and that the 
results obtained are not indefinite or approximate, but are certain and exact. 



EXAMPLES. 

1. Assuming the result in Art. 4 (6), namely, tnat the slope of the tangent 
at a point (x\, y{) on the curve y = x 2 is 2x\, find the slope and the angle 
made with the x-axis by the tangent at each of the points whose abscissas are 

.5, 0, 1, 1.5, 2, 2.5, 3, 4, -2, -3, - i, - f, - f . 

2. In the curve in Ex. 1 find the coordinates of the points the tangents at 
which make angles of 20°, 30°, 45°, 60°, 85°, 115°, 145°, 160°, 170°, respec- 
tively, with the x-axis. 

Av 

3. Draw figures of the following curves. Find the value of — - at any 

A? 
point (x, y) in the case of each curve ; then find what -~- is approaching 

when Ax approaches zero : 

(a) x 2 + y 2 = 16; (6) Jf = a? + x + l; (c) y = x* ; 

(d) y 2 = $x; (e) 9 x 2 + 16 if = 144 ; (/) 9 x 2 - 16 y* = 144 ; 

(gr) yi=4px; (h) b 2 x 2 + a 2 ?/ 2 = « 2 5 2 ; (i) 6 2 x 2 - ahf = a°-b 2 . 

* The mathematical phraseology and notation employed to express these 
ideas is given in Chapter II. 



10 



BIFFEREN TIAL CALCULUS. 



[Ch. I. 



^Suggestion. In (a), (x + Ax) 2 + (y + A?/) 2 = 16. It can then be de- 

L Ay 2 x + Ax "1 

ducedthat Ax = ~2iTA^J 

Compare the results found in (g), (h), and (i), with those found in 
analytic geometry. 

4. Using the results obtained in Ex. 3, find the slopes and the angles made 
with the x-axis by the tangents in the following cases : 

(a) The curve in Ex. 3 (a), at the points whose abscissas are 

4, 2, 1, 0, - 1.5, -3.5. 

(6) The curve in Ex. 3 (c), at the points whose abscissas are 

-3, -2,-1, 0, 1.5, 2.5. 

(c) The curve in Ex. 3 ((f), at the points whose abscissas are 

0, 1, 2, 3, 6, 8. 

(d) The curve in Ex. 3 (e), at the points whose abscissas are 

0, 1, 2, 4, -.5, -1.5. 

(e) The curve in Ex. 3 (/), at the points whose abscissas are 

4, 8, 10, -5, -7. 

5. Using the results obtained in Ex. 3, find the points on the curve in 
Ex. 3 (a) the tangents at which make angles 40° and 136° with the x-axis. 

6. Do as in Ex. 5 for the curves whose equations are given in Ex. 3 (c), 
(d), 00, and (/). 

7. Do some of the examples in Art. 62. Make careful drawings in each 



5. To determine the area of a plane figure. A plane area, say 
ABCD, may be supposed to be divided into an exceedingly great 

number of exceedingly small rect- 
angles. It will be seen later 
that the limit of the sum of these 
rectangles when they are taken 
smaller and smaller, is the area. 
The calculus furnishes a way to 
find this limit. Even at this 

stage in the study of the calculus 
Fig. 2. 

the student can get some useful 

ideas concerning this problem by making a brief inspection of 

Art. 165, Exs. (a), (6), (c). [Art. 14 discusses the term "limit."] 




5-7.] INTRODUCTORY PROBLEMS. 11 

6. (a) To find a function when its rate of change at any (every) 
moment is known, or, in more general terms, when its law of change 
is known. In Art. 3 (b) a particular example has been given of 
this general problem, viz. to determine the rate of change of a func- 
tion at any moment. The calculus not only provides a method of 
solving this general problem, but also provides a method of solving 
the inverse problem which is stated above. 

(b) To find the equation of a curve when its slope at any (every) 
point is known. In Art. 4 (b) a particular example has been given 
of this general problem, viz. to determine the slope of a curve at 
any point on it. The calculus not only provides a method of 
solving this problem, but it also provides a method of solving the 
inverse problem which has just been stated. Problem (b) is a 
special case of problem (a), for the slope at a point on a curve 
really shows "the law of change" existing between the ordinate 
and the abscissa of the point (see Art. 26). 

A brief inspection of Arts. 24-26, 167,169, at this time, will repay 
the beginner. 

Note. Differential calculus and integral calculus. The subject of 
infinitesimal calculus is frequently divided into two parts ; namely, differential 
calculus and integral calculus. This division is merely a formal division ; 
though oftentimes convenient, it is by no means necessary. Examples of the 
kind given in Arts. 2-4 formally belong to "the differential calculus," and 
those described in Arts. 5, 6, to "the integral calculus." 

7. Elementary notions used in infinitesimal calculus. The prob- 
lems used in Arts. 2-4 put in evidence some notions and methods, 
the consideration and development of which constitute an impor- 
tant part of infinitesimal calculus. These notions are : 

(1) The notion of varying quantities which may approach as 
near to zero as one pleases, such as A^ and Aa^ in the last stages 
of the solution of the problems in Arts. 3 and 4. 

(2) The notion of a varying quantity, such as — - 1 in Art. 3 

/or -^ in Art. 4 J, which approaches a fixed number when A^ 

(or Ax x ) varies and decreases towards zero, and approaches in such 
a way that the difference between the varying quantity and the 
fixed number can be made to become, and remain, as small as one 
pleases, merely by decreasing A^ (or Ax x ). 



12 DIFFERENTIAL CALCULUS. [Ch. I. 

The infinitesimal calculus gives mathematical definiteness and 
exactness to these notions, and a convenient notation has been 
invented for dealing with them. From these notions, with the 
help of this notation, it has developed methods and obtained 
results which are of great service in such widely separated fields 
of study as geometry, astronomy, physics, mechanics, geology, 
chemistry, and political economy. 

A review of certain notions of algebra is not only highly advan- 
tageous but absolutely necessary for a satisfactory understanding 
of the calculus and for good progress in its study. Accordingly, 
Chapter II. is devoted to the consideration of the notions of a 
variable, a function, a limit, and continuity. 

Note. Reference for collateral reading. Perry, Calculus for Engi- 
neers, Preface, and Arts. 1-18. 



CHAPTER II. 

ALGEBRAIC NOTIONS WHICH ARE FREQUENTLY 
USED IN THE CALCULUS. 

8. Variables. When in the course of an investigation a quan- 
tity can take different values, the quantity is called a variable 
quantity, or, briefly, a variable. For instance, in the example in 
Art. 3, the distance through which the body falls and its speed 
both vary from moment to moment, and, accordingly, are said to 
be variables. Again, if the x in the expression x 2 -f- 3 be allowed 
to take various values, then x is said to be a variable, and x 2 -f 3 
is likewise a variable. If a steamer is going from New York to 
Liverpool, its distance from either port is a variable. 

In general a variable can take an unlimited number of values. 

Note 1. Numbers. The values of a variable are indicated by numbers. 
In preceding mathematical work various kinds of numbers have been met ; 
such as 2, 7, f, V2, y/b, ir = 3.14159 ••-, log 10 8 = .90309 •••, e = 2.71828 •••, 
V— 5, 3 V— 1, 4 + 3 V— 1. The student is supposed to be acquainted 
with the divisions of numbers into real and imaginary, integral and frac- 
tional, rational and irrational, positive and negative. In general in this 
book real numbers only are used. 

Graphical representation of real numbers. Draw a straight line LM, 

L C Q AD BG M 

1 X 1 1 1 l 

-i 1 vT~ 3VT0 

Fig. 3. 
which is supposed to be unlimited in length both to the right and to the 
left. Choose any point 0, and take any distance OA for unit length. Also 
let it be arranged for convenience (as has been done in trigonometry and 
analytic geometry) that positive numbers be measured from towards M, 
and negative numbers from towards L. Then the point A represents the 
number 1 ; if OB = 3 OA, B represents the number 3 ; if OC = \ OA, C 
represents the number — J. If OD is the length of a diagonal of a square 
whose side is OA, then OD = V2, and D represents the number V2 ; if OG 
be the length of a diagonal of a rectangle whose sides are OA and OB, then 
OG = VlO, and G represents the number VlO. It is a topic for a more ad- 
vanced course than this to show that all real numbers can be represented on 

13 



14 DIFFERENTIAL CALCULUS. [Ch. IL 

the unlimited line LM, that to each point on LM there corresponds (on the 
scale OA = 1) a definite real number, and that to each real number there 
corresponds a definite point on the line. 

Absolute value of a number. The value of a number without regard to 
sign is called its absolute value. Thus the absolute values of the numbers 
1, — 2, -|, — i are 1, 2, |, i. The absolute value of a number x is denoted 
by the symbol \x\. 

Note 2. Infinite numbers. Sometimes the value of a variable " be- 
comes unlimited in magnitude, 1 ' i.e. "increases beyond all bounds." The 
variable is then said to become infinite in magnitude, and its value is then 
called infinity. If the unlimited value is positive, it is denoted by the 
symbol + oo ; if it is negative, it is denoted by the symbol — oo. For ex- 
ample, if x be an angle, as x increases from 45° to 90°, tan x increases from 
-j- 1 to + oo ; and as x decreases from 135° to 90,° tanx decreases from — 1 
to — oo. 

The symbol oo does not denote a definite number in the same way as 2, 
say, denotes a number ; the symbol oo merely means that the measure of the 
variable concerned is unlimitedly great, or, in other words, is beyond all 
bounds.* 

9. Functions. When two variables are so related that the value 
of one of them depends upon the value of the other, each is said to be 
a function of the other. 

For example, the area of a circle depends upon the length of its radius, 
and so the area is said to be a function of the radius. To a definite value of 
the radius, e.g. 2 inches, there corresponds a definite value of the area, viz. 
irx2 2 inches, i.e. 12.57 sq. in. 

Another example : the length of the side of a square depends upon the 
area of the square, and so the side is said to be a function of the area. To a 
definite value of the area, say 9 sq. in., there corresponds a definite side, viz., 
a side 3 inches in length. 

The idea of a function is sometimes expressed thus : When 
two variables are so related that to any arbitrarily assigned definite 
value of one of them there corresponds a definite value (or set of 
definite values) of the other, the second variable is said to be a 
function of the first.~\ 

* For further notes on numbers, and especially for references for reading, 
see Infinitesimal Calculus, Art. 8. Additional references are Pierpont, 
Theory of Functions of Real Variables, Chaps. I., II. ; Veblen-Lennes, Infini- 
tesimal Analysis, Chaps. I., II., and the references given on pages 10, 11, 19. 

t See Veblen-Lennes, Infinitesimal Analysis, Chap. III. (and its historical 
note on page 44). 



9.] FUNCTIONS. 15 

For example, suppose y = x 2 + 2 x — 5. (1) 

When the value 3 is assigned to x, y must take the corresponding value 
3 2 + 2 x 3 — 5, i.e. 10 ; when x is — 2, y must be — 5. In these cases y is 
said to be a function of x ; also x is called the independent variable and y is 
called the dependent variable. 

On the other hand when the value 30 is assigned to y, x must have the 
corresponding values 5 and — 7. (These values are obtained by substituting 
30 for y in (1), and then solving for x.) When y is 115, x must be 10 or 
— 12. In these cases x is said to be a function of y ; also y is called the 
independent variable, and x is called the dependent variable. 

Ex. Given that x 2 - y 2 - 6 x-Sy -7 = 0: (2) 

(a) assign values to x and find the corresponding values of y ; 

(b) assign values to y and find the corresponding values of x. 
Independent variable; dependent variable. The variable which 

can take arbitrarily assigned values is usually termed the inde- 
pendent variable; the other variable, whose values must then be 
determined in order that they may correspond to these assigned 
values, is usually termed the dependent variable. It is evident 
that if the second definition above be followed, " function " and 
"dependent variable" are synonymous terms. 

One-valned functions. Many-valued functions. When a function 
has only one value corresponding to each value of the independent 
variable, the function is called a one-valued function ; when it has 
two values it is called a tico-valued function. If a function has several 
values corresponding to each value of the independent variable, 
it is called a multiple-valued function, or a many-valued function. 

For example: In (1), y is a one-valued function of x, and x is a two- 
valued function of y. If y = x 2 , y is a single-valued function of x ; if y = Vx, 
y is a two-valued function of x. 

If y = sin x, y is a one-valued function of x. 

If y = sin -1 x, i.e. (using another notation) if y = arc sinx,* y is a many- 
valued function of x. 

Inverse functions. If y is a function of x, then, on the other 
hand, x is a function of y. The second function x is called the 
inverse function of the first function y. That is, if 

y =/(«), (3) 

then x = 4>(y), ( 4 ) 

* See Plane Trigonometry, Arts. 17, 88. 



16 DIFFERENTIAL CALCULUS. [Ch. II. 

in which. <f>(y) denotes an expression in y which is obtained by 
solving equation (3) for x. 

E.g. in (1), y = x 2 + 2 x - 5. 

On solving for x, there is obtained the inverse function, 



x = - 1 ± Vy + Q. 

Again, if y — a x , the inverse function is x = log a y ; if y = sin x, the inverse 
function is x = sin- 1 y ; or as it is frequently written x = arc sin y. 

Functions of two variables. Functions of more than two variables. 

The value of a function may depend upOn the values assigned to 
two or more other variables. In such a case the first variable is 
said to be a function of the other two variables. 

E.g. If z = x 2 + y 2 + 18, z is said to be a function of x and y ; 
if v = u 2 + w 2 + t 2 + 5, v is a function of u, w, and t. 

10. Constants. A quantity whose value never changes through- 
out an investigation is called a constant. 

If a constant remains the same in all investigations, it is called 
an absolute constant. 

Thus 2, .33, ir, are absolute constants. 

A quantity which has a fixed value in one investigation and 
another fixed value in another investigation is called an arbitrary 
constant. 

Thus let the equations of a straight line, (x, y) denoting any 
point on the line, be 

y — mx + b and x cos a -f- y sin a =j). 

Here m and b, a and p, are arbitrary constants. For any partic- 
ular line a and p have fixed particular values, and so also have 
m and b. 

11. Classification of Functions. 

A. Explicit and implicit functions. When a function is expressed 
directly in terms of the dependent variable, like y in equation (1), 
Art. 9, the function is said to be an explicit function. When 
the function is not so expressed, as in equation (2), Art. 9, it is 
said to be an implicit function. If relation (2), Art. 9, were solved 
for y, then y would be expressed as an explicit function of x ; thus 



10, 11.] CLASSIFICATION OF FUNCTIONS. 17 

On solving the same relation for x, the variable x is expressed 
as an explicit function of y\ thus 

*=±(y + 4) + 3. 

B. Algebraic and transcendental functions. Functions may also 
be classified according to the operations involved in the relation 
connecting a function and its dependent variable (or variables). 
When the relation involves only a finite number of terms, and 
the variables are affected only by the operations of addition, sub- 
traction, multiplication, division, raising of powers, and extraction 
of roots, the function is said to be algebraic; in all other cases 

it is said to be transcendental. Thus 2 x 2 -f- 3 x — 7, V^ + -, are 

x 

algebraic functions of x; sin x, tan (# + «)? cos -1 a, l x , e 2x , logic, 
log 3 a, are transcendental functions of x. The elementary tran- 
scendental functions are the trigonometric, anti-trigonometric, ex- 
ponential, and logarithmic. Examples of these have just been 
given. 

C. Rational and irrational functions. Algebraic functions are 
subdivided into rational functions and irrational functions. Ex- 
pressions involving x which consist of a finite number of terms 
of the form ax 11 , in which a is a constant and n a positive integer, 

e.g. 3 x 4 — 2 X s + 4 x + 5, 

are called rational integral functions of x. 

When these expressions have more than two terms they are 
also called polynom ials in x. 

If an expression in x, in which x has positive integral expo- 
nents only, and which has a finite number of terms, includes 
division by a rational integral function of x, 

x-1 

e ' g ' , , or + i x — 2, 

oar+i 6 xr -f 9 

it is called a rational fractional function of x. 

Rational integral functions and rational fractional functions 
are included together in the term rational functions. 



18 DIFFERENTIAL CALCULUS. [Ch. II. 

An expression which involves root extraction of terms involv- 
ing x is called an irrational function of x ; 



e.g. Vx, -Vx 2 + 3sc + 5 + 9a; — 2. 

D. Continuous and discontinuous functions. A discussion on this 
exceedingly important classification of functions is contained in 
Art, 16. 

12. Notation. In general discussions variables are usually 
denoted by the last letters of the alphabet, x, y, z, u, v, •••, and 
constants by the first letters, a, b, c, ■••. 

The mere fact that a quantity is a function of a single variable, 
x, say, is indicated by writing the function in one of the forms 
f(x), F(x), cj>(x), '•■,fi(x),f 2 (x), •••. If one of these occurs alone, 
it is read " a function of x " or " some function of x " ; if several 
are together, they are read " the /-function of x" " the F-i unction 
of x," "the phi-function of x" •••. The letter y is often used to 
denote a function of x. 

The fact that a quantity is a function of several variables, 
x, y, z, •••, say, is indicated by denoting the quantity by means of 
some one of the symbols, f(x, y), <f>(x, y), F(x, y, z), if/(x, y, z, u), •••. 
These are read " the /-function of x and y," " the phi-function of 
x and y" " the F-i unction of x, y, and z," etc. 

Sometimes the exact relation between the function and the 
dependent variable (or variables) is stated; as, for example, 

f(x) =x 2 + 3x — 7,ory = x 2 + 3x — 7; F(x, y) = 2 e x + 7 e y + xy - 1. 

In such, cases the /-function of any other number is obtained by 
substituting this number for x in f(x), and the F-f unction of any 
two numbers is obtained by substituting them for x and y respec- 
tively in F(x, y). Thus 

f(z) =z 2 + 3z-7, /(4) = 4 2 + 3-4-7 = 21; 
F(t, z) = 2 e* + 7 e* + tz - 1, F(2, 3) = 2 e 2 + 7 e 3 + 5. 

In a way the phrases "expression containing x " and "function of x" 
may be regarded as synonymous. In finding the value of an explicit func- 
tion corresponding to a particular value of the variable, the expression in- 
volving the variable is treated simply as a pattern form in which to substitute 
the value of the variable. 



12, 13.] GRAPHICAL BEPBESEXTAriON OF FUNCTIONS, 19 

EXAMPLES. 

1. Calculate /(2) and /(. 1) when /(«) = 8 Vx 4- - + 7 x 2 + 2. Write f(y) , 
/(m),/(sinx)/ x 

2. Calculate /(2, 3), /(-2, 1), and /(- 1, -1) when /(*, y) = 
3 x 2 + 4 xy + 7 y- - 13 x + 2 y - 11. Write /(w, v), /(sin sc, 2). 

9 J. Qy 

3. Calculate s as a function of x when y = f(x) = — — — and s =f(&. 

4 — 7 x 

4. Given that f(x) = x* + 2 and i^x) = 4 + Va, calculate /[.?(«)] and 

5. Jif(x, y) = «x 2 + bxy + c*/ 2 , write /(y, x), /(*, »), and/(?/, ?/). 

6. If y = f(x) = ax + h , show that x=f(y\ 

ex — a 

7. If y = (t>(x) = " x ~ — , show that x = <(>(y), and that x = 2 (x), in 

3 x — 2 

which 2 (x) is used to denote 0[0(x)]. 

8. If /(x) = x + 1 , show that / 2 (x)=x, f 4 (x) = x, /«(&)= x, etc., in 

x — 1 

which / 2 (x) is used to denote /[/(x)], / :3 (x) to denote /{/[/(x)]}, etc. 



If /(x) = ■=! , show that /W-ZCy) = * 



x+1 l+/0*0-/Q/) 1+X0 

Xote. Notation for inverse functions. The student is already familiar 
with the trigonometric functions and their inverse functions, and with the 
notation employed ; thus, y = tan x, and x = tan -1 y. In general if y is a 
function of x, say y = /(x) , then x is a function of y. The latter is often 
expressed thus : x =f~ 1 (y). For instance, if y = log x, x = log- 1 (ij). This 
notation was explained in England first by J. F. TV. Herschell in 1813, and at 
an earlier date in Germany by an analyst named Burmann. See Herschell, 
A Collection of Examples of the Application of the Calculus of Finite 
Differences (Cambridge, 1820), page 5, note. 

13. Graphical representation of functions of one variable. This 
topic is discussed in algebra and in analytic geometry. 

For instance, if y = 7 x + o, (1) 

the line whose equation is (1) is the graph of the function y in (1). 

If x 2 + y 2 = 25, (2) 

the circle whose equation is (2) is the graph of the function y in 
(2). Important properties of a function can sometimes be in- 



20 DIFFERENTIAL CALCULUS. [Ch. II. 

ferred or deduced from an inspection of its graph.* Illustrations 
of this will appear in later articles. 

14. Limits. The notion that varying quantities may have fixed 
limiting values is very important and should be clearly understood 
when the study of the calculus is entered npon. 

Limit of a variable. When a variable y, say, on taking successive 
values approaches nearer and nearer to a constant value a, in such 
a way that the absolute value of the difference between y and a be- 
comes and remains less than any preassigned positive quantity, the 
constant a is said to be the limit of the variable y 9 and y is said to 
approach the limit a, 

EXAMPLES. 

1. The area of a regular polygon inscribed in a circle varies when the 
number of its sides is increased. Also, this area then approaches nearer 
and nearer to the area of the circle. Further, the difference between the 
area of the circle and the area of the polygon with the increasing number of 
sides can be made less than any quantity that may be arbitrarily assigned, 
simply by increasing the number of the sides. Moreover, this difference re- 
mains less than the arbitrarily assigned quantity, when the number of sides 
is still further increased. 

This is mathematically expressed thus : 

" The limit of the area of a regular polygon inscribed in a circle, when 
the number of sides is increasing beyond all bounds, is the area of the circle ; " 
and also expressed thus : 

" The area of the polygon approaches the area of the circle as a limit when 
the number of its sides is increasing beyond all bounds." 

(In this case the varying polygonal area is always less than its limit, the 
area of the circle.) 

2. Discuss the case of the area of the regular circumscribing polygon when 
the number of its sides is continually increasing. 

(In this case the varying polygonal area is always greater than its limit.) 

3. Discuss the cases of the lengths of the varying perimeters of the poly- 
gons in Exs. 1, 2. 

4. The number — , in which n is a positive integer, decreases as n in- 

2 n 

creases, and its value approaches nearer and nearer to zero when n is increased. 



* Not every function can be represented by a curve ; see Infinitesimal 
Calculus, page 20, footnote. 



14. J LIMITS. 21 

Also, — can be made to differ from zero by as small a positive Dumber as 

may be assigned, simply by increasing n ; and the difference between — and 

zero continues to remain less than the assigned number when n is still further 
increased. 

Accordingly, — approaches zero as limit, when n becomes unlimitedly 
great. In other words : 

the limiting value of — , for n increasing beyond all bounds, is zero. 

5. Let S n denote the sum of n terms of the geometric series 

2 4 2 n_1 

The first term is 1 ; the sum of the first two terms is 1| ; the sum of the first 
three terms is 1| ; the sum of the first four terms is Iff ; and so on. It thus 
seems to be the case that the more terms are taken, the nearer is their sum 
to 2. This is clearly evident on writing the sum of n terms ; for 

I - 1 2»-i 

Accordingly (see Ex. 4), S n approaches 2 as limit when n increases be- 
yond all bounds ; 
in other words : 

the limiting value of the series 1 + | + \ + •••, the number of whose terms is 
unlimited, is 2. 

N.B. The following trigonometric examples of limits are important, and 
will be employed in later articles. Proofs of 6, 7, 8, are given in text-books 
on trigonometry. 

6. (a) When an angle is approaching 0° the limiting value of sin0 is 0. 
(6) When angle is approaching 90° the limit of sin is 1. 

(c) When angle is approaching 0° the limit of cos0 is 1. 

{d) When angle is approaching 90° the limit of cos 6 is 0. 

(e) When angle 6 is approaching 0° the limit of tan is 0. 

(/) When angle is approaching 90° tan becomes unlimitedly great. 

7. Show that, being the number of radians in the angle, the limiting 

value of the fraction , when is approaching zero, is unity. 



In Fig. 4, angle AOP = radians ; QBE is a circular arc described about 

as centre with radius r ; QMB is a chord drawn at right angles to OA, 



22 



DIFFERENTIAL CALCULUS. 



[Ch. II. 



and accordingly is bisected by OA at M ; $!T and BT are tangents drawn 
at Q and i?, which must meet at some point T on OA. 




Fig. 4. 

By trigonometry, MQ = rsinfl, arc QB = rd, QT = rtan 

By geometry, chord QB < arc QBB < broken line QTB ; 

z.e. 2MQ<2a,rcBQ<2 QT. 

. : , from (1) , 2 r sin 6<2r0<2 r tan ^. 

sin^<^<tan^. 

1 



sin 6 1 cos 



(1) 



(2) 

(3) 



. • . , on division by sin $, 
Now let d approach zero. 

From the fact in Ex. 6 (c), the limit of is then 1. 

v J ' COS0 

a 

Accordingly, since by relation (3), the value of lies between 1 and 

sin# 

a 

a number which is approaching 1 as its limit, the limit of must also be 

sin0 



1. Hence, the limit of 



sin 9 



when 6 is approaching zero, is 1.* 



8. Show that the limiting value of 



tan 6 



is 1 when Q approaches zero. 



[Suggestion. Divide the quantities in relation 2, Ex. 7, by tan0.] 
9. Show that the limit of , when x approaches a, is 2 a. 



x — a 
10. Show that the limit of the sum 2 — 1 
increases beyond all bounds, is f. 



— ••• to n terms, as n 



For another proof see Plane Trigonometry, pages 143, 144. 



15.] NOTATION. 23 

11. In Ex. (a), Art. 4, — ^ varies with Ax, and approaches 4 as Ax 

Ax 

approaches zero. By decreasing Ax the difference between =^ and 4 can be 

Ax 

made less than any positive number that may be assigned, and will remain 
less than this number when Ax continues to decrease. That is, the limit of 

— ^, as Ax approaches zero, is 4. 
Ax 

Show that in Ex. (6), Art. 4, the limit of — ^, as Ax approaches zero, is 2 x. 

Ax 

Xote 1 . In each of these cases — ^ finally reaches its limit. In Ex. 10 

Ax 

the variable sum can never reach its limit. 

As 

12. In Ex. (5), Art. 3, — varies with At, and approaches gt as At 

At 

As 
approaches zero. By decreasing At the difference between — and gt can be 

At 

made less than any positive number that may be assigned, and will remain 

less than this number when At continues to decrease. Accordingly, the limit 

As 

of — , as At approaches zero, is gt. 
At 

As 
In Ex. (a), Art. 3, the limit of — , as At approaches zero, is 128.8. 

At 
As 
In each of these cases — can reach its limit. 
At 

Another form of the definition of a limit. In the following defini- 
tion, which, is longer than the preceding one, the circumstances 
under which the dependent variable approaches a limit are expli- 
citly expressed. 

Definition of a limit. Let there be a function of a variable, and 
let the variable approach a particular value. If, at the same time 
as the variable approaches the particular value, the function also 
approaches a fixed constant in such a way that the absolute value 
of the difference between the function and the constant may be made 
less than any positive number that may be assigned; and if, more- 
over, this difference continues to remain less than the assigned num- 
ber ivhen the variable approaches still nearer to the particular value 
chosen for it; then the constant is the limit of the function when the 
variable approaches the particular value. 

Ex. Read Exs. 1-12, with this definition in mind. 

15. Notation. The limit of a variable quantity, and the con- 
dition under which this limit is approached, are expressed by 



24 DIFFERENTIAL CALCULUS. [Ch. II. 

means of a certain mathematical shorthand. Thus the last sen- 
tence in Ex. 5, Art. 14, is expressed : 

Lim^ 00 (l+i + i+ -) = 2. 

The results found in Ex. 11 are expressed : 

Lim A ^ ^ = 4; Lim A ^ ^ = 2a. 

Ax Ax 

The result found, in Ex. 6 (b) is expressed : 
Lim^zr sin = 1. 

"— 2 

The symbol = is placed between a variable and a constant in 
order to indicate that the variable approaches the constant as a 

limit. Thus 6 = ^ above, means that approaches ^ as a limit. 

Note. The symbol = is used to indicate an approach to equality. The 
symbol = is used by many instead of = to indicate the same idea. Various 
other notations are also employed. 

Ex. Express the results in Exs. 1-12 in the mathematical manner of 

writing. 

15 a. Continuous variation. Interval of variation. When a vari- 
able number, x say, takes in succession in the order of their mag- 
nitudes all values from a number a to a number b, x is said to 
vary continuously from a to b. The set of numbers from a to & 
constitute what is called the interval from a to 6, and this interval 
is denoted by [a, b~\ or by (a, b)* 

The notion of a variable that varies continuously through an 
interval [a, b] may be described graphically. 

V A P B_ 

~~ ] a x b 

Fig. 5.f 

On this line let the distances be measured from 0, OA = a, and 
OB = b. The point A thus corresponds to the number a, and the 



* This symbol should not be confounded with a similar symbol which has 
an altogether different meaning, the symbol denoting a point in analytic 
geometry. 

t The point may happen to be between A and B or may be to the right 
of B. 



16.] CONTINUOUS FUNCTIONS. 25 

point B to the number b. Let P be any point on the segment 
AB, and x its corresponding number. Then as the point P 
moves along the line from A to B, it passes in succession through 
all the points from A to B ; and thus its corresponding number x 
takes for its successive values all numbers, in the order of their 
magnitudes, from a to b. 

16. Continuous functions. Discontinuous functions. A function 
f(x) is said to be continuous for the value x = c if it satisfies both the 
following conditions : 

(1) Its value is finite when x = c, i.e. f(c) is finite; 

(2) The difference f{c 4- h) —f(c) approaches zero as the abso- 
lute value of h approaches zero. 

If, in the case of a function f(x), either of the conditions (1) and 
(2) is not fulfilled when x has a particular value, say x = c, then the 
function f(x) is said to be discontinuous for the value x = c, or, more 
briefly, discontinuous at c. 

A function f(x) is said to vary continuously from a to h 9 or to be 
continuous in the interval (a, 6), # ivhen it is continuous for every 
value of x between a and b. 

The last definition may be written more fully on making use of 
the first : 

A function f(x) is said to be. a continuous function of x for all 

values of x from x = a to x = b, if it satisfies the following 
conditions : 

(1) Its value is finite for all values of x between a and b ; 

(2) Any two numbers between a and b (say c and c -f h) being 
taken, the difference f(c + h)— f(c) approaches zero as the abso- 
lute value of h approaches zero. 

Note 1. Condition (2) may be roughly expressed in the following way, 
which helps to bring out its practical meaning : 

The change made in /(x) is exceedingly small when an exceedingly small 
change is made in cc, while the value of x lies between a and b. Or, in other 
words, the value of /(x) does not take a sudden jump of either a finite or an 
unlimited amount when x changes by only an exceedingly small amount at 
any value between a and b. 

* See Art. 15 a. 



26 DIFFERENTIAL CALCULUS. [Ch. II. 

EXAMPLES. 

1. Let /(x) = x 2 + 3x-7. 

This function is finite for all finite values of x ; accordingly, /(x) satisfies 
condition (1) for any finite values of a and b. 

Let Xi and x\ + ft be any finite values of x. Then 

f(xi) = xJ + Sxi-7,, 

and /(xi + h) = (x x + h)' 2 + 3(aci + ft) - 7. 

Hence, the difference /(xi + ft) -/(xi) = ft (2 Xi + ft + 3). 

This difference approaches zero when ft approaches zero ; accordingly, 
f(x) satisfies condition (2). 

Since x 2 + 3 x — 7 thus satisfies conditions (1) and (2) , it is continuous for 
all finite values of x. 

This example may be made more concrete by giving xi a value, 3 say. 

Then /(3 + h) - /(S) = 9 h + h 2 , 

which approaches zero when h approaches zero. 

.-. /(&) is continuous for x = 3. 

2. Show that the function is continuous for values of x from — 4 to 

x — 1 

+ i, and for values of x from f to 5. 

3. Show that the function, f(x) = , is discontinuous when x = 1. 

Give x the value 1 + h. 

1 1 



Then/(l + fc) = 



(1 + 70-1 ft 



The value of /(I + ft) evidently increases beyond all bounds when ft ap- 
proaches zero. Thus f(x) does not satisfy condition (1) when x = 1 ; and, 
accordingly, is discontinuous for the value x = 1. 

Note 1. Further examination shows that when x is passing through the 

value 1, is going through an unlimitedly great change in value. 

x — 1 

When 2 is a little less than 1, say .99999, then -J— = =— 1000000. 

J x — 1 .99999-1 

When x is a little more than 1, say 1.000001, then -^— = — — £- — - = + 1.000000. 

a; — 1 1.000001 — 1 

The difference between the values of x here is 1.000001 — .99999, i.e. 

.000002 ; the difference between the corresponding values of the function is 

1000000 -(- 1000000), i.e. 2000000. 



16.] CONTINUOUS FUNCTIONS. 27 

In general : 

When £ is a little less than 1, say 1 — h, in which h is a very small number, 

then -J- = 1 =-!; 

x-1 (1-/0-1 h 

when £ is a little more than 1, say 1 + h, 

then -1- = I =1. 

x - 1 (1 + 7i) - 1 /i 

Accordingly, /(l + A) - f(l - h) = 2 • 

The smaller 7i is made, the greater this difference becomes ; and it in- 
creases beyond all bounds when 7i approaches zero. Thus experiences 

x — 1 

an unlimited change in value when x passes through the value 1. 



that when x passes through the value — , tana; takes an unlimitedly great 

change in value. 

Note 2. Some functions experience finite changes in value when the 
variable passes through particular values. 

For example : 

l i 

the function f(x) = 2 (4*-* -l)~ (4?-8 + 1) 

changes its value from — 2 to + 2 (i.e. by the amount 4) when x is passing 
through the value 3.* 

Note 3. References for collateral reading on Limits and Continuous 
and Discontinuous Functions. Several are given in Infinitesimal Calculus, 
p. 29 ; to these add Pierpont, Theory of Functions of Beat Variables, Vol. L, 
Chap. YL, VII., Veblen-Lennes, Infinitesimal Analysis, Chaps. IV., V. 



* See Infinitesimal Calculus, pages 26-29, Exs. 3, 6, Notes 5, 6, 9. 



CHAPTER III. 

INFINITESIMALS, DERIVATIVES, DIFFERENTIALS, 
ANTI-DERIVATIVES, AND ANTI-DIFFERENTIALS. 

17. In this chapter some of the principal terms used in the 
calculus are denned and discussed, and one of the main problems 
of the calculus is described. In the first study of the calculus 
it is better, perhaps, not to read all this chapter very closely, 
but after a cursory reading of it to proceed to Chapter IV., and, 
while working the examples in that chapter, to re-read carefully 
the articles of this chapter. These articles can also be reviewed 
most profitably when the special problems to which they are 
applied are taken up. Articles 22, 23, however, should be care- 
fully studied before Chapter IV. is begun. 

18. Infinitesimals, infinite numbers, finite numbers. An infini- 
tesimal is a variable which has zero for its limit. (See definition 
of a limit, Art. 14.) That is, if a denote an infinitesimal, 

a = 0, or limit a = 0. 

Tor instance, in Ex. (a), Art. 4, when PR is approaching zero it 
is an infinitesimal. So also, at the same time, are angle QPT 
and the triangle PQR. Again, when angle 6 is an infinitesimal 
sin and tan are infinitesimal ; cos is an infinitesimal when 

is approaching ^ ; when n is increasing beyond all bounds 1 -s- 2 n 

is an infinitesimal. 

Note. The infinitesimal of the calculus is not the same as the infinitesimal 
of ordinary speech. The latter is popularly defined as "an exceedingly small 
quantity," and is usually understood to have a fixed value. The infinitesimal 
of the calculus, on the other hand, is a variable which approaches zero in a 
particular way. 

28 



17-19.] INFINITESIMALS. 29 

The following statements are in accordance with, or follow 
directly from, the definitions of a limit and an infinitesimal. 

(1) The difference between a variable and its limit is an 
infinitesimal. That is, on denoting the variable by x and the 
limit by a, 

if limit x = a, i.e. if x = a, 

then x = a + a, in which a = 0. 

(2) If the difference between a constant and a variable is an 
infinitesimal, then the constant is the limit of the variable. In 
symbols, if x = a + a, 

in which a = 0, 

then x = a, 

i.e. limit x = a. 

This principle has been employed in the exercises in Arts. 3, 4. 

It is evident that the reciprocal of an infinitesimal approaches 
a nnmber which is greater than any number that can be named, 
namely, an infinite number. Accordingly, an infinite number may 
be defined as the reciprocal of an infinitesimal. Numbers which 
are neither infinitesimal nor infinite are called finite numbers. 

19. Orders of magnitude. Orders of infinitesimals. Orders of 
infinites. Let m and n each denote a number which may be 
finite, infinite, or infinitesimal. When the limiting value of the 

ratio — is a finite number, m and n are said to be finite with 

n 
respect to each other and to be of the same order of magnitude; 

when the ratio _ either has the limit zero or is beyond all bounds, 
n 

m and n are said to be of different orders of magnitude. 

For instance, 1,897,000,000 and .000001 are of the same order of magni- 
tude. Tan 90° and tan 45° are of different orders of magnitude. Logx 
and x are of different orders of magnitude when x is an infinite number. 
This is shown in Art. 118, Ex. 1. 

That infinitesimals may be of different orders of magnitude is 
shown by the following illustration. 



30 



DIFFERENTIA L CALC UL US. 



[Ch. III. 



Suppose that the edge BL of the cube in Fig. 6 is divided into any number 
of parts, and that each part, as Bb, becomes infinitesimal. Through each 
point of division, as b, let planes be passed at right angles to BL. The 
cube is thereby divided into an infinite number of 
infinitesimal slices like Bd. Now suppose that the 
edge BA is divided like BL into parts like Bf which 
become infinitesimal, and let a plane be passed 
through each point of division / at right angles to BA. 
The slice Bd is thereby divided into an infinite num- 
ber of infinitesimal parallelopipeds like Ck. Finally 
suppose that the edge BO is divided into parts which 
become infinitesimal like Bg, and that through each 
point of division, as g, a plane is passed at right 
angles to BC. Then Ck is thereby divided into an 
infinite number of infinitesimal parallelopipeds like 

kg. Since the limiting value of each of the ratios , — -, — -, is infinite, 

Bd Ck kg 

the parallelopipeds DL, Bd, Ck, kg, are all of different orders of magnitude. 
This illustration also serves to show that infinites may be of 
different orders of magnitude. 




Each of the three ratios, 



, — -, — , is an infinite number. But the 

kg kg kg 

ratio of the first to the second, viz., : — , i.e. - — is an infinite num- 

kg kg Bd 

ber; accordingly the first and second ratios are of different orders of magni- 
tude. Similarly it can be shown that the second and third ratios are of 
different orders of magnitude. 

Note. On infinitesimals see Infinitesimal Calculus, pages 32-38, espe- 
cially the References, page 38. 

20. Changes or increments in the variable and the function. 
A, Change in the variable. Suppose that 

and that x has a particular value, say x x . Then y has a particu- 
lar value, viz. y 1 ^=f( K x 1 ). 

Now suppose that x changes from x x by a certain amount, which 
may be denoted by Aas. 

This symbol Ax — which is read ' delta-x 1 (see Art. 3, Note 1) — 
means simply a change or difference made in the value of x. 
This change, which may be either an increase or a decrease, is 
often called ^ ie i UC rement of the variable oc. 



20, 21.] CHANGE IN THE FUNCTION. 31 

Increment of x (i.e. Ax) = (new value of x) — (the old value of x). 
E.g. if x changes from the value 4 to 4.2, 

its increment = 4.2 — 4, or .2 ; i.e. Ax = .2. 
If x changes from the value 4 to 3.6, 

its increment = 3.6 — 4, or — .4 ; i.e. Ax = — .4. 

B, Change in the function. When a variable x changes, its 
function y changes and, accordingly, has an increment. This incre- 
ment is denoted by Ay, 

Thus Ay = (new value of y) — (old value of y). 

E.g. let y = 3x 2 -4x + 5. 

If x = 4, then y = 3 x 4 2 - 4 x 4 + 5 = 37. (a) 

Let x receive an increment Ax = .2. 

Then ?/ receives an increment Ay, and the new value of y, viz. , 

y + Ay = 3 x (4.2) 2 - 4 x (4.2) + 5 = 41.12. (6) 

.*. On subtraction in (a) and (6), Ay = 41.12 - 37 = 4.12. (c) 

In general : if y =f(x), (1) 

and x receives an increment Ax, 
then y also receives an increment Ay. 
Then (1) becomes y + ky =f(x + Ax), (2) 

and, thus, from (1) and (2), Ay =f(x + Ax) -f(x). (3) 

In accordance with the use of the symbol A, the second mem- 
ber of (3) may be written &f(x). 

EXAMPLES. 

1 . Given y = x 3 — 3 x + 4, calculate the corresponding increment of y, i.e. 
Ay, when : 

(a) x = 5 and Ax = .3 ; (b) x = 3 and Ax = — .2. 

2. Given s = 32 1 2 + 17 t — 5, calculate the corresponding increment of s, 
i.e. As, when: 

(a) * = 3 and At = .1 ; (6) £ = 6 and At = .1 ; (c) « = 8 and At = .3. 

3. Given r = sin 0, find the increment of r, i.e. Ar, when : 
(a) = 37°, Ad = 20' • (6) = 216°, A0 = 1°. 

4. Given r = cos 0, find the increment of r, when : 
(a) = 37°, Ad = 20' ; (b) 6 = 216°, A0 = 1°. 

5. See tables of results on pages 3, 6, for examples on increments. 

21. Comparison of the corresponding changes (or increments) made 
in a function and the variable. These increments are compared by 
forming the ratio, incre ment of the fanction § 
increment of the variable " 



32 DIFFERENTIAL CALCULUS. Ch. III. 

That is, if the function is denoted by f(x), (1) 

by forming the ratio /(x + ^~ /( *° - (2) 

The fraction expressed by the form (2) is called the 
difference-quotient of the function. 

EXAMPLES. 

1. In the example worked in Art. 20, JB, in which 

Ax = .2, and the corresponding Ay = 4.12, 

^> = 1^ = 20.6. 

Ax .2 

2. See last columns of tables, pages 3, 6, for examples of comparison of 
increments. 

3. Calculate the difference-quotients — in Ex. 1, Art. 20. 

Ax 

As 

4. Calculate the difference-quotients — in Ex. 2, Art. 20. 

5. Calculate the difference-quotients — in Exs. 3, 4, Art. 20. 

AS 

22. The derivative of a function of one variable. Suppose that 
the function f(x) 

denotes a continuous function of x. Let x receive an increment 
Ax ; then the function becomes 

/(x + Ax). (a) 

Hence the corresponding increment of the function is 

/(x + Ax)-/(x). (b) 

This may be written A [/(a;)]. 

The ratio of this increment of the function to the increment of 
the variable is f( x + Ax ) __f(x) ^ _ A[/(a?)] a (c) 

Ax ' ' ' Ax 

The limit of this ratio when Ax approaches zero, i.e. 

.. f(x + Ax)-f(x) .. A/(x) , 7 , 

llm ^~ &x ° r hmA ^ AaT' W 

is caWed the derived function of f(x) with respect to x; or the 
derivative (or the derivate) of f(x) with respect to x; or the 
^■derivative of /(as). It is also called £/ie differential coefficient 
off(x), a name which is explained in Art. 27. 



22.] DIFFEBENTIATION. 33 

If y also be used to denote the function, that is, if 

y =/O0, 

then if x receive an increment Ax, y will receive a corresponding increment 
(positive or negative) , which may be denoted by Ay, i. e. 

y + Ay =f(x+ Ax). 

Hence Ay = f(x + Ax) - /(x) ; 

and *» = /(s + **)-/(s) . (e) 

Ax Ax w 

.-. lim A ^ = lim A ^ /^ + Ax)-/(x) . 

Ax Ax 

The process of finding the derivative of a function is called 
differentiation. This process is a perfectly general one, as indi- 
cated in steps (a), {b), (c), and id). It may be described in 
words, thus : 

(1) Give the independent variable an increment ; 

(2) Find the corresponding increment of the function ; 

(3) Write the ratio of the increment of the function to the 
increment of the variable. 

(4) Find the limit of this ratio as the increment of the variable 
approaches zero. 

For a slightly different description of the process of differentiation, see 
Note 4. 

Note 1. To differentiate a function {i.e. to find its derivative) is one 
of the three main problems of the infinitesimal calculus, and is the main 
problem of that branch which is called " the differential calculus.'''' 

Note 2. The other two main problems of the infinitesimal calculus (see 
Arts. 27 «, 164) are the main problems of that branch called " the integral 
calculus.'''' It may be said here that while the differential calculus solves the 
problem, "when the function is given, to find the derivative," on the other 
hand the integral calculus solves as one of its two main problems the inverse 
problem, namely, "when the derivative is given, to find the function." 

EXAMPLES. 

1. Find the derivative of x 3 with respect to x. 

Here /(x) = x 3 . (See Fig., p.462.) 

Let x receive an increment Ax ; 
then f(x + Ax) = (x + Ax) 3 = x 3 + 3 x 2 Ax + 3 x(Ax) 2 + (Ax) 3 . 



34 DIFFERENTIAL CALCULUS, [Ch. III. 

.-. f(x + Ax) - f(x) = 3 x 2 Ax + 3 x (Ax) 2 + (Ax) 3 . 

... /(* + Ax)-/(x) = 3 x , + 3 xAx + (Aa . )2> 
Ax 

.•■Um^.g ' + *">-* *> = 8*. 

Ax 

If ?/ be used to denote the function, thus y = x 3 , then the first members of 
these equations will be successively, y, y + Ay, Ay, -^, lim Axi0 -^ 

AX AX 

Note 3. It should be observed that the expression (c) depends both on 
the value of x and the value of Ax, and, in general, contains terms that 
vanish with Ax, as exemplified in Ex. 1. (This is shown clearly in Art. 150.) 
On the other hand, the value of the derivative depends on the value which 
x has when it receives the increment, and on that alone. For this reason, the 
derivative of a function is often called the derived function. For instance, 
in Ex. 1, if x = 2, the value of the derivative is 12 ; if x = 6, the value of 
the derivative is 108. Compare Exs. in Arts. 3, 4. (It is probably now 
apparent to the beginner that the process used in the problems in Arts. 3, 4, 
was nothing more or less than differentiation.) 

Note 4. Sometimes Ax is called the difference of the variable x, (b) is 
called the corresponding difference of the function, and (c) is called the 
difference-quotient of the function. The process of differentiation may then 
be described, thus : (1) Make a difference in the independent variable ; 
(2) Calculate the corresponding difference made in the function ; (3) Write 
the ratio of the difference in the function to the difference in the variable ; 
(4) Determine the limiting value of this ratio when the difference in the 
variable approaches zero as a limit. 

2. Find the derivatives, with respect to x, of x, 2 x, 3x, ax, x 2 , 7x 2 , 
11 x 2 , 6x 2 , x 3 , 5 x 3 , 13 x 3 , and ex 3 . 

Ans. 1, 2, 3, a, 2x, 14 x, 22 x, 2&x, 3x 2 , 15 x 2 , 39 x 2 , 3 ex 2 . 

3. Calculate the values of these functions and the values of their 
derivatives, when x = 1, x = 2, x = 3. 

4. Find the derivatives, with respect to x, of : (a) x 2 + 2, x 2 — 7, 
x 2 + k ; (b) x 3 + 7, x 3 - 9, x 3 -f c. 

1 2 

5. Differentiate x 4 , x 2 + 4 x — 5, -, — 3 x + 2 x 2 , with respect to x. 

6. Find the derivatives, with respect to t, of 3 f-, 4 t 3 - 8 t + -• 

3 7 

7. Differentiate y G , -y 2 — 8y — , with respect to y. 

4 y 

8. Show that, if n is a positive integer, the derivative of x n with respect 
to x, is wx»-l. 

Note 5. The result in Ex. 8» as will be seen later, is true for all con- 
stant values of n. 



23.] NOTATION. 35 

9. Assuming the result in Ex. 8, apply it to solve Exs. 4-7. 

Xote 6. In order that a function may be differentiable (i.e. have a de- 
rivative), it must be continuous ; all continuous functions, however, are not 
differentiable. Eor remarks on this topic, see Echols, Calculus, Art. 30. 
For an example of a continuous function which has nowhere a determinate 
derivative, see Echols, Calculus, Appendix, Xote 1, or Harkness and Morley, 
Theory of Functions, § 65 ; also Pierpont, Functions, Vol. I., Arts. 367-371. 

23. Notation. There are various ways of indicating the deriva- 
tive of a function of a single variable. (In what follows, the 
independent variable is denoted by x. In the case of other 
variables the symbols are similar to those now to be described 
for functions of x.) 

(a) This symbol is often used to denote (d) Art. 22, viz. 

/'(«). A 

Thus the derivatives (or derived functions) of F(x), $(?/), f(t), 
fxiz), with respect to x, y, t, and z, respectively, are denoted by 
F(x), cf>'(i/), f'(t), fi(z). These are sometimes read " the i^-prime 
function of %," etc. 

(&) If y is used to denote the function of x (see Art. 22), the 
derivative of y with respect to x is frequently indicated by the 
symbol f< B 

This is often read " y-prime " ; but it is better to say " deriva- 
tive of y." 

(c) The ^-derivative of f(x) is also indicated by the symbol 

i-iW C;or by M. 

The brackets in D are usually omitted, and the symbol is written 

df(x) 



dsc 



E 



Symbols C, D, and E should be read "the a>derivative of f(x). v 
(<f) When y denotes the function, the derivative (see Equation 
(/) Art. 22) is sometimes denoted by 



36 DIFFERENTIAL CALCULUS. [Ch. III. 

The brackets in F and G are usually omitted, and the symbol 
for the derivative is written 

dx 

This should be read for a while at least by beginners, "the 
derivative of y with respect to x" or more briefly " the x-derivative 
°fy" (Other phrases, e.g. " dy by dx," are common, but, unfortu- 
nately, are misleading.) 

(e) In case (d) the operation of differentiation, and also its 
result, namely, the derivative, are alike indicated by the symbol 

Dy. I 

(f) Sometimes the independent variable x is shown in the 
symbol, thus D y, T 

Note 1. Mathematics deals with various notions, and it discusses these 
notions in a language of its own. In the study of any branch of mathe- 
matics, the student has first to clearly understand its fundamental notions, 
and then to learn the peculiar shorthand language, made up of signs and 
symbols and phrases, which has been in part invented, and in part adapted, 
by mathematicians. A striking instance of the great importance of mere 
notation is seen in arithmetic. To-day a young pupil can easily perform 
arithmetical operations which would have taxed the powers of the great 
Greek mathematicians. The one enjoys the advantage of the convenient 
Arabic notation* for numerals, the other was hampered by the clumsy 
notation of the Greeks. 

Note 2. Symbols A and B, and also /and J", have this important quality, 
namely, they tend to make manifest the fact that the derivative is a single 
quantity. It is not the ratio of two things, but is the limiting value of a 
variable ratio. Symbols C and F have the quality that they indicate, in a way, 

the process (Art. 22) by which the derivative is obtained. The symbol — 

dx 
before a function indicates that the operation of differentiation with respect 
to x is to be performed on the function ; it also serves to indicate the result 
of the operation. The symbols D and 2? x ,t in /and J, are simply abbrevia- 
tions for the symbol — • 
dx 

* This should really be called the Hindoo notation ; for the Arabs obtained 
it from the Hindoos. See Cajori, History of Mathematics. 

t The symbol D x y is due to Louis Arbogaste (1759-1803), professor of 
mathematics at Strasburg. The symbol -^ was devised by Leibnitz, and 
the symbol /', by Lagrange (1736-1813). dx 



24.] 



REPRESENTATION OF THE DERIVATIVE. 



Note 8. Beginners in the calculus are liable to be misled by the symbols 

Z>, E, G, and H, especially by H. The symbol -^ does not denote a fraction : 

dx 
it does not mean "the ratio of a quantity dy to a quantity dx." Such quan- 

dy 

dx 

thoroughly realized, and never forgotten, that -^ is short for — (y), and 

dx dx 

that both these symbols are merely abbreviations for lim^^o — (see ~Eq.(f) 

Art. 22). Some one has remarked that the dy and dx in -^ are merely " the 

dx 
ghosts of departed quantities" ; but perhaps this is claiming too much for 
them. 



24. The geometrical meaning and representation of the derivative 
of a function. Let f(x) denote a function, and let the geometrical 
representation of the function, namely the curve 



be drawn. 



y=/(»)i 



(1) 

















&/ 








Y 


P^ 




5 
^y 


f 


s 7 












8 i 


a 








Ail 


-\ 










' 


h 
























-^L 


/T 





^-Kt-+ 












X 



Fig. 7. 



Let P(x 1 , 2/j) and Qfa + Ax^ yi + Ay^) be two points on the 
curve. Draw the secant LPQ. Then 



tan XLP = 



A?/! 

Aoj, 



Now let secant LQ revolve about P until Q reaches P. Then 
the secant LP takes the position of the tangent TP, and the 
angle PLX becomes PTX ; then, also, Axj reaches zero. 



Hence 



tan XTP = lim Azi=0 



&x. 



(2) 



38 DIFFERENTIAL CALCULUS. [Ch. III. 

Now P (»!, 2/j) is any point on the curve ; hence, on letting 
(x, y), according to the usual custom, denote any point on the 
curve, and <f> denote the angle made with the a>axis by the 
tangent at (x, y), 

tan<£ = lim^^|. (3) 

The first member of (3) is the slope of the tangent at any point 
(x, y) on the curve y—f(x), and the second member is the 

derivative of either member of (1). Hence -^, i.e, /'(a?), is the 
slope of the tangent at any point (oc, y) on the curve y = /(a?). 

This principle has already been applied in the exercises in 
Art. 4. 

Curve of slopes. If the graph of f'(x) be drawn, that is, the 
curve y=f'(x), it is called the curve of slopes of the curve 
y —f{x). It is also called the derived curve, and sometimes the 
differential curve of y ==/(#). For instance, the curve of slopes 
of the curve y = x 2 is the line y = 2 x. The curve of slopes is 
the geometrical representative of the derivative of the function ; 
the measure of any of its ordinates is the same as the slope of 
y = f(x) for the same value of x. 

Ex. Sketch the graphs of the functions in Exs., Art. 22. Write the 
equations of these graphs. Give the equations of their curves of slopes, and 
sketch these curves. (Use the same axes for a curve and its curve of slopes.) 

Note 1. Produce RQ (Fig. 7) to meet TP in S, produce PR to R', and 
draw R'Q'S' parallel to R Q to meet the curve in Q' and TP in S'. Then 



Now, if A Xl = PR, ^ = fi ; and if A Xl = PR', %r = W ' Als0 



dz v '. PB rs 1 






g-*g, -«*.-«'. 


Ayi_ 
Axi 


B'Q' 

' PR' 


,. BQ dy 
hm ^pl = dx' 






,. B'Q 1 dy 
llm ^FW = di- 







and likewise, 

Note 2. Hereafter, in general investigations like the above, the symbol x 
will be used instead of xi to denote any particular value of x ; and similarly 
in the case of other variables. 



25.] MEANING OF THE DERIVATIVE. 39 

25. The physical meaning of the derivative of a function. Sup- 
pose that the value of a function, say s, depends upon time ; 
i.e. suppose s=f(t). 

After an interval of time At, the function receives an incre- 
ment As; and . . „,. , AjN 
s + As=f(t + At). 

.-. As=f(t + At)-f(f). 



A8 = f(t + At)-f(f) 

At At 



(1) 



r.\im^Mi.e.^)=f'(t). 



At \ dt y 

As 

Since As is the change in the function during the time At-, — 

is the average rate of change of the function during that time. 
As At decreases, the average rate of change becomes more nearly 
equal to the rate of change at the time t, and can be made to 
differ from it by as little as one pleases, merely by decreasing At. 
Hence the second member of (2) is the actual rate of change 
at the time t. In words : The derivative of a function with respect 
to the time is the rate of change of the function. 

CtS 

If s denotes a varying distance along a straight line, then — 
denotes the rate of change of this distance, i.e. a velocity. 

(For discussions on speed and velocity see text-books on Kine- 
matics and Dynamics, and Mechanics.) 

Ex. Show that if s = \ gt 2 , then — = gt. (See Art. 3 6.) 

Note. Newton called the calculus the Method of Fluxions. Variahle 
quantities were called by him fluents or flowing quantities, and the rate of 
flow, i.e. the rate of increase of a variable, he called the fluxion of the 

fluent. Thus, if s and x are variable, — and — are their fluxions. Newton 

dt dt 

indicated these fluxions thus : s, x. This notation was adopted in England 
and held complete sway there until early in the last century, and the other 
notation, that of Leibnitz, prevailed on the continent. At last the continental 
notation was accepted in England. " The British began to deplore the very 
small progress that science was making in England as compared with its 
racing progress on the continent. In 1813 the ' Analytical Society ' was 
formed at Cambridge. This was a small club established by George Peacock, 



40 DIFFERENTIAL CALCULUS. [Ch. III. 

John Herschel, Charles Babbage, and a few other Cambridge students, to 
promote, as it was humorously expressed, the principles of pure ' D-ism,' 
that is, the Leibnitzian notation in the calculus against those of ' dot-age,' 
or of the Newtonian notation. The struggle ended in the introduction into 

Cambridge of the notation -^, to the exclusion of the fluxional notation y. 

dx 
This was a great step in advance, not on account of any great superiority of 
the Leibnitzian over the Newtonian notation, but because the adoption of the 
former opened up to English students the vast storehouses of continental 
discoveries. Sir William Thomson, Tait, and some other modern writers 
find it frequently convenient to use both notations." — Cajori, History of 
Mathematics, page 283. 

26. General meaning of the derivative : the derivative is a rate. 

When a variable changes, a function of the variable also changes. 
A comparison of the change in the function with the causal change 
in the variable will determine the rate of change of the function 
with respect to the variable. The limit of the result of this com- 
parison, as the change in the variable approaches zero, evidently 
gives this rate. But this limit has been defined as the derivative 
of the function with respect to the variable. Accordingly (see 
Art. 22, Note 1), the main object of the differential calculus may be 
said to be the determination of the rate of change of the function 
with respect to its argument. 

Note 1. The rate of change of the function with respect to the variable 
may also be shown in a manner that explicitly involves the notion of time. 
In the case of the function y, when y = f(x), let it be supposed that x receives 
a change Ax in a certain finite time At. Accordingly y will receive a change 
Ay in the same time At. Then, from the equation preceding (e), Art. 22, 

Ay _ fix + Ax) - f(x) _ f(x + Ax) — f(x) _ Ax _ ,. 

At At Ax " At' 

Assume that Ax =£ when At ^= 0. When At approaches zero, Ax also 
approaches zero. On letting At approach zero, and writing the consequent 
limits of the three fractions in (a), there is obtained , 

^=f<(x) d *-,i.eM = %L.^. (1) Whence, §1^ (2) 

dt dt dt dx dt dx dx v J 

Result (2) can also be derived directly from 

Ay 

^ = ~ (P) 

Ax Ax v J 

At 



26.] DIFFERENTIALS. 41 

(Here it is assumed that Ax # 0, when At =£ 0.) When At approaches 
zero, Ax approaches zero. On letting At approach zero, and writing the con- 
sequent limits of the three fractions in (&), relation (2) is obtained, and 
from it relation (1) follows. 

Ex. Express relations (1) and (2) in icords. 

Thus the derivative of a function with respect to a variable may be regarded 
as the ratio of the rate of change of the function to the rate of change of the 
variable. 

Note 2. References for collateral reading. McMahon and Snyder, 
Diff. Cal., Arts. 88, 89 ; Lamb, Calculus, Art. 33 ; Gibson, Calculus, Arts. 
31-37, 51. 

EXAMPLES. 

1. A square plate of metal is expanding under the action of heat, and 
its side is increasing at a uniform rate of .01 inch per hour; what is the 
rate of increase of the area of the plate at the moment when the side is 16 
inches long ? At what rate is the area increasing 10 hours later ? 

Let x denote the side of the square and A denote its area. Then A = x 2 . 

Now ±A = ±A . 4* whence? dA = dA . to , m *A =2 xx .01 sq. inches 

At Ax At dt dx dt dt 

per hour = .02 x sq. inches per hour. Accordingly, at the moment when the 
side is 16 inches, the area of the plate is increasing at the rate of .32 sq. inches 
per hour. Ten hours later the side is 16. 1 inches ; the area of the plate is 
then increasing at the rate of .322 sq. inches per hour. The area of the 
square is increasing in square inches 2 x times as fast as the side is increasing 
in linear inches. 

2. In the case of a circular plate expanding under the action of heat, 
the area is increasing at any instant how many times as fast as the radius ? 
If when the radius is 8 inches it is increasing .03 inches per second, at what 
rate is the area increasing ? At what rate is the area increasing when the 
radius is 15 inches long ? 

3. The area of an equilateral triangle is expanding how many times as 
fast as each of its sides ? At what rate is the area increasing when each 
side is 15 inches long and increasing at the rate of 2 inches a second ? At 
what rate is the area increasing when each side is 30 inches long and increas- 
ing at the rate of 2 inches a second ? 

4. The volume of a spherical soap bubble is increasing how many times as 
fast as its radius ? At what rate (cubic inches per second) is the volume in- 
creasing when the radius is half an inch and increasing at the rate of 3 inches 
per second ? At what rate is the volume increasing when the radius is an inch ? 

5. A man 5 ft. 10 in. high walks directly away from an electric light 16 
feet high at the rate of 3| miles per hour. How fast does the end of his 
shadow move along the pavement ? 



42 DIFFERENTIAL CALCULUS. [Ch. III. 

27. Differentials, (a) Differential of a variable. 

Let an independent variable x have a change Ax. 
This difference Ax in x is often called 

' the differential ofx'-, 

and it is then customary to denote it by the symbol 

dx. (1) 

(6) Differential of a function. 

Let/(V) denote any differentiate function. 
Its derivative (Art. 23) is denoted by f'(x). 

The product of the derivative of a function f(x) and the differen- 
tial of the independent variable, viz. 

f'(x)dx (2) 

is called the differential off(x). 

In the same fashion as the differential of a variable x is denoted 
by dx, the differentials of any other variables u, v, w, y, • ••, are 
denoted by du, dv, dw, dy, •••. 

Now let y denote the function f(x) ; i.e. 

On taking the derivatives, — = f'(x). (3) 

ax 

Then, by the definitions and notation above, 

dy=f'(x)dx; (4) 

i.e. dy = ^ . dx. (5) 

The defining equations (4) and (5) may be expressed in words : 

The differential of a function y of an independent variable x is 
equal to the derivative of the function multiplied by the differential 
of the variable, the latter differential being merely a change (or dif- 
ference) made in the variable. 



27.] DIFFEBENTIALS. 43 

The letter d is used as the symbol for the differential. 
E.g. the differential of f(x) is written df(x). 
Thus, by definition (b), 

df(x) = f(x)dx. 

Illustration : If y = ^, 

then $y = 3x 2 . 

dx 

. •. dy = -^- • dx = 3 # 2 dx. 
dx 

If # = 4, and d# = .01, 

cfy = 3 x 4 2 x .01. 

= .48. 

The actual change made in y when x changes from 4 to 4.01 is 

(4.01) 3 -4 3 = . 481201. 

It will be found that, as in this case, the differential of a func- 
tion corresponding to an assigned differential of the variable is 
not in general the same as the change in the function ; it is, how- 
ever, approximately equal to this change. 

Note 1. The differential dx of an independent variable x may be any 
arbitrary change, usually small, or it may be an infinitesimal. In the exam- 
ples in this article the differentials have arbitrarily assigned or determinable 
values ; in the examples, in the integral calculus the differentials employed 
are usually infinitesimals. 

Note 2. It is highly important to notice that in Equations (3) and (4), 

dy and dx are used in altogether different ways.* In (3) and (5), -^ is used 

dx 

as a symbol for lim^x^ — ; and it denotes the definite limiting value of a 
Ax 

difference-quotient. In (4) and in (5) on the extreme right dx is not zero 

(although it may happen to be, and usually is, a small quantity),! and the 

dy is such that the ratio dy : dx is equal to f'(x). For instance, in Fig. 7, 

* In one respect this double use of dx and dy is unfortunate ; for it tends 
to confuse beginners in calculus. Other notation is also used. 

t Later on many examples will be found in which this dx is an infinitesimal. 



44 DIFFERENTIAL CALCULUS. [Ch. III. 

y - of Equation (2) is tan SPR. As to Equations (4), (5), if dx = PR, then dy 

= RS, and if dx= PR', then dy = R'S'. This shows that dy, in (4), is the 
increment of the ordinate of the tangent corresponding to an increment dx 
of the abscissa. The corresponding increment of the ordinate of the curve 
y=f(x) [i.e. the increment of the function /(x)] in some cases can he 
found exactly by means of the equation of the curve, and in some cases can 
be found, in general only approximately, by means of a very important 
theorem in the calculus, namely, Taylor's Theorem (see Chap. XVI.). 
Instances of the former are given below ; instances of the latter are given 
in Art. 150. 

Note 2. It should be clearly understood that, according to the preceding 
remarks, cancellation of the dx's in (5) is impossible. 

N.B. For geometric illustrations of derivatives and differentials see 
Art. 67. 

EXAMPLES. 

1. In the case of a falling body s = \ gt 2 (see Art. 3) ; on denoting, as 
usual, the differential of the time by dt, ds, the corresponding differential of 
the distance is [Ex., Art. 3 (&)] gtdt ; i.e. ds = gldt. The actual change in s 
corresponding to the change dt in the time is [see Eq. (2), Art. 3 (&)] 
gtdt + \g(dt) 2 . 

2. In the curve y = x 2 , dy = 2 x dx. The actual change in y corresponding 
to the change dx in x is 2 x dx -f (dx)' 2 . (See Eq. (1), Art. 4.) Thus if x = 10 
and dx = .001, dy = 2 x 10 x .001 = .02. The actual change in the ordinate of 
the curve from x = 10 to x = 10 + .001 is (10.001) 2 ~ 10 2 , i.e. .020001. This 
change may also be calculated as stated above, viz. 2 x 10 x .001 + (.001) 2 . The 
dy = .02 is the change in the ordinate of the tangent at x = 10 from x = 10 to 
x = 10.001 (see Note 1). (The student should use a figure with this example.) 

3. Write the differentials of the functions in the Exs. in Art. 22. 

4. Given that y = x 3 — 4 x 2 , find dy when x = 4 and dx = .1. Then find 
the change made in y when x changes from 4 to 4.1. 

5. Given that ?/ = 2 x 3 + 7 x 2 — 9 x + 5, find dy when x = 5 and dx = .2. 
Then find the change made in y when x changes from 5 to 5.2. 

Note 3. It is evident from these examples that the differential of a 
function is an approximation to the change in the function caused by 
a differential change in the variable ; and that the smaller the differential 
of the variable, the closer is the approximation. When the differential varies 
and approaches zero it becomes an infinitesimal. 

Ex. Calculate the differentials of the areas in Ex. 2, Art. 26, when the 
differential of the radius is .1 inch. 

Ex. Calculate the differentials of the areas of the triangles in Ex. 3, 
Art. 26, when the differential of the side is .1 inch. 



27a.] ANTI-DERIVATIVES AND ANTI-DIFFEBENTIALS. 45 

Note 4. It may be remarked here that in problems involving the use 
of the differential calculus derivatives more frequently occur, and in prob- 
lems in integral calculus differentials (viz. infinitesimal differentials) are 
more in evidence. 

Note 5. Keferences for collateral reading. Gibson, Calculus, § 60 ; 
Lamb, Calculus, Arts. 57, 58. 

27 a. Anti-derivatives and anti-differentials. In Arts. 22 and 27 
the derivative and the differential of a function have been defined, 
and a general method of deducing them from the function has 
been described. With respect to the derivative and the differen- 
tial the function is called an anti-derivative and an anti-differential 
respectively. Thus, if the function is x 2 , the ^-derivative and the 
^differential are 2 x and 2 xdx respectively ; on the other hand, 
x 2 is said to be an anti-derivative of 2 x and an anti-differential of 
2 xdx. To find the anti-derivatives and the anti-differentials of a 
given expression is one of the two main problems of the integral 
calculus. (See Art. 22, Notes 1, 2, and Arts. 164, 166, 167.) 

Note. Reference for collateral reading. Perry, Calculus for Engi- 
neers, Arts. 12-24, 28, 66. 



CHAPTER IV. 

DIFFERENTIATION OF THE ORDINARY FUNCTIONS. 

28. In this chapter the derivatives of the ordinary functions of 
elementary mathematics are obtained by the fundamental and 
general method described in Art. 22. Since these derivatives are 
frequently employed, a ready knowledge of them will prevent 
stumbling and thus make the subsequent work in calculus much 
simpler and easier; just as a ready command of the sums and 
products of a few numbers facilitates arithmetical work. Accord- 
ingly these derivatives should be tabulated by the student and 
memorized. 

N.B. The beginner is earnestly recommended to try to derive these results 
for himself. For a synopsis of the chapter see Table of Contents. 

GENERAL RESULTS IN DIFFERENTIATION. 

29. The derivative of the sum of a function and a constant, namely, 

Put. y = <j>(x) + c. 

Let x receive an increment Ax; consequently y receives an 
increment, Ay say. That is, 

y + Ay = <f> (x + Ax) + c. 

.-. Ay = cf>(x + Ax) + c - [<£<» + c] 

= <f) (x -f Aoj) — <£ (x). 

Ay _ <ft (x + Ax) — <fr (x) 
Ax Ax 

46 



29.] 



DIFFERENTIATION OF FUXCTIOXS. 



47 



Let A.r approach zero as a limit ; then 



lira 



A* 



lira 



<ft (s + A.r)- <£(*) . 



Ax 



i.e. 



i.e. 






(i) 



Hence, (f constant terms appear in a function, they may be neg- 
lected ichen the function is differentiated. 

If u be nsed to denote <f>(x), result (1) can be expressed: 



£<«+•>=£ 



(2) 



Cok. 1. It follows from (1) that the derivative of a constant is 

zero. This may also be derived thus: If y — c a constant, then 

y + \y = c\ and, accordingly, \y = 0. Hence, — -^ = for all 

values of Ax*; hence, -^, i.e. —(c), is zero. 
dx dx 

Cok. 2. If two functions differ by a constant, they have the 
same derivative. 

From (2) and Art. 27, d(u + c) = du. 

Xote 1. In geometry y = c is the equation of a straight line parallel to the 
axis of x and at a distance c from it. The slope of this line is zero ; this is in 
accord with Cor. 1. 

Note 2. The curves y = 4>(x) + c, in which c is an arbitrary constant 
(Art. 10), can be obtained by moving the curve y = 4>(x) in a direction 
parallel to the ?/-axis. The result (1) shows that for the same value of the 

abscissa, the slope -&■ is the same for all the curves. See Figs. 8, 9, below. 
dx 




Fig. 8. 



48 DIFFERENTIAL CALCULUS. [Ch. IV. 

Note 3. The converse of Cor. 1 is also true ; namely, if the derivative of 
a quantity is zero, the quantity is a constant. 

Ex. Show this geometrically. (See Art. 24.) 

Note 4. The converse of Cor. 2 is also true ; namely, if two functions 
have the same derivative, the functions differ only by an arbitrary constant. 
(By the same derivative is meant the same expression in the variable and the 
fixed constants.) For let 4>(x) and F(x) denote the functions, and put 

y = <p{x)~ F(x). 

By hypothesis, Dy = <p'(x) - F'(x) = 0. 

Hence, by Note 3, y — c ; 

and accordingly, <p(x) = F(x) + c. 

Ex. Show this geometrically. 

Note 5. If -^ = </>'(x), then y = (j>(x) + c, in which c denotes any con- 
dx 
stant. Hence <p(x) + c is a general expression for all the functions whose 
derivatives are 4>'(x). Functions such as <f>(x) + 1, (p(x) — 3, obtained by 
giving particular values to c, are particular functions having the same deriva- 
tive 4>'(x). 

Note 6. Notes 4 and 5 come to this : The anti-deriyatiye of a function 
is indefinite, so far as an arbitrary additive constant is concerned. 

30. The derivative of the product of a constant and a function, say 
c<Kas). 

Put y = ccf>(x). 

Let x receive an increment Ax; consequently y receives an 
increment, Ay say. 

That is, y + Ay = ccf> (x + Ax) . 

.-. Ay = c[<f>(x + Aaf) — <£(#)]. 

' ' Ax 



= r<f>(x + Ax)-<l>(x) l 

L A * J 

}\ m A ^-lim „[ <Ka + Aa;)-<Ka?)l i 



i.e. JL= c <p'(x); 

ax 

i.e. J-[c<K*)] = c4>'(a;). (1) 



30, 31.] DIFFERENTIATION OF FUNCTIONS. 49 

That is, the derivative of the product of a constant and a function 
is the product of the constant and the derivative of the function. 

If </> (x) be denoted by u, then (1) is written 

£™= c %\ ( 2 ) 

In particular, if u = x, — (ex) = c. 
dx 

From the above and the definition in Art. 27, <2[c<£(x)] = 
cc? [<£(#)], d(cu) = cdu, d(cx) = cdx. 

Ex. See Exs., Art. 22. 

31. The derivative of the sum of a finite number of functions, say 

<j>0*0 + F(oc) + •••. 

Put y=<J>(x)+F(x) + ~.. 

Then, on giving x an increment Ax (as in Arts. 29, 30), 

y + Ay = <£(» + Ax) + 2^(aj + Ax) + •••• 

.-. Ay = <f>(x + Ax) - <f>(x) + i^(x + Ax) - F(x) + 

. Ay = <f>(x + Ax) — <fr(x) i^(x + Ax) - F(x) _ 
Ax Ax Ax 

Hence, on letting Ax approach zero, 

<k = A < jJx) + —F(x) + •••; (1) 

dx dx^ K } dx w ' w 

i.e. A [<f,( x ) + ^(a-) + -]=*'(»).+ *"(*) + • • •■ ( 2 ) 

That is, £fte derivative of a sum of a finite number of functions 
is the sum of their derivatives. 

If the functions be denoted by u, v, w, • • •, i.e. if 

y = u + v + w-\ — , 

the result (1) may be expressed thus : 

dy _ du dv . dw , 

doc doc dx doc 



50 DIFFERENTIAL CALCULUS. [Ch. IV. 

From this and Art. 27, 

dy = du + dv + dw + ... . 

Note 1. The differentiation of the sum of an infinite number of functions 
is discussed in Art. 147. 

In working the following exercise the result of Ex. 8, Art. 22, may be 
used. 

Ex. Find the derivatives of 

2 x* + 7 x 2 - 10 x + 11, x 2 - 17 x + 10, - x 2 + 21 x - 5. 

32. The derivative of the product of two functions, say $(x)F(x). 

Put y = 4>(x)F(x). 

Then, on giving x an increment Ax, 

y + Ay = <f>(x + Ax)F(x + Asc). 

.-. Ay = <£(a; + Ax)F(x + Ax) - <fi(x)F(x). 

a . m Ay = <ft(a? + Ax)F(x + Ax) - cj>(x)F(x) (± 

Ax Ax ^ ' 

On letting Ax = 0, the second member approaches the form - • 

In order to evaluate this form, introduce <{>(x + Ax)F(x) — cf>(x-\- 
Ax)F(x) in the numerator of this member.* Then, on combining 
and arranging terms, (1) becomes 

Ax Ax Ax 

Hence, on letting Ax approach zero, 

f> = 4>(x)F'(x) + F(x)<j>'(x). (2) 

U.X 

That is : The derivative of the product of two functions is equal to 
the product of the first by the derivative of the second pins the 
product of the second by the derivative of the first. 



* Equally well, tf>(x) F(x + Ax) — <p(x) F(x + Ax) may be thus introduced. 
The student should do this as an exercise. 



32.] DIFFERENTIATION OF FUNCTIONS. 51 

If the functions be denoted by u and v, that is, if 
y = uv, 
then (2) may be expressed 

dx dx dx y 

The derivative of the product of any finite number of functions 
can be obtained by an extension of (3). For example, if 

y = uvw, 

then, on regarding vw as a single function, 

dy f sdu , d / n 

-^ = (viv) \- u — (vie) 

dx v J dx dx K J 



du . 

= vw \-u[ IV 

dx 



( dv . dw\ 

IV \-v ) 

\ dx dx ) 



du , dv . div /A , 

= vw h ivu f- u v — • (4) 

dx dx dx 

Similarly, if y = uvivz, 

di/ du , dv , dw , dz /KN 

-^ = vivz — ■ + invz h uvz h uvw — (5) 

dx dx dx dx dx 

In general : In order to find the derivative of a product of several 
functions, multiply the derivative of each function in turn by all 
the other functions, and add the results. 

Xote. Another way of obtaining (5) is given in Art. 39 (a). 

The differential of the product of two functions. If 

y = uv, 
then, from (3) and the definition in Art. 27, it follows that 

du = u — dx + v- — dx. (6) 

9 dx dx W 

But, by Art. 27, —dx = dv, and —dx = du. 
dx dx 

Hence, (6) may be written 

d(uv) = udv -f vdu, (7 X ' 



52 DIFFERENTIAL CALCULUS. [Ch. IV. 

Similarly, if y = www, 

it follows from (4) that dy = vwdu + wmcfa; + uvdw. 

On division by uvw, this takes the form 

d (uvw) du dv dw /0 , 
= 1 1 (8) 

UVW uvw 

Ex. 1. Write dy in forms (7) and (8), when y = uvwz. 

Ex. 2. Differentiate O 3 + l)(x 2 - 2 a: + 7) by the above method ; then 
expand this product and differentiate, and show that the results are the 
same. 

Ex. 3. Treat the following functions as indicated in Ex. 2 : 

x 2 (x - 1) (x s + 4), (ax 2 + bx + c)(lx + m). 

Ex. 4. Write the differentials of the functions in Exs. 2, 3. 

33. The derivative of the quotient of two functions, say <K«0 ■*■ F(x). 

Put y = ±M. 

F(x) 

Then, on proceeding as in Arts. 29-32, 

Fix + Ax) 

. A?/ = <frfo + Ax) <ft(s) 
" * i^(x + Ax) F(x) 

= <fr(x + Ax)g(x) - cj>(x)F(x + Ax) 
"' ~" F(x)F(x + Ax) 

. Ay __ <fr(x + Ax)F(x) — <f>(x)F(x + Ax) ~. 

" Ax F(x)F(x + Ax) Ax ^ ' 

On letting Ax = 0, the second member approaches the form - • 
In order to evaluate this form, introduce 

F(x)<f>(x) - ^(*)*'(o0 

in the numerator of this member. Then, on combining and 
arranging terms, (1) becomes 

F(x) U(x + Ax)-<f>(x)l _ V F(x + As) - F(x) l 

Ay_ W L Ax J yv ; L Ax J 



Ax F(x)F(x + Ax) 



33.] DIFFERENTIATION OF FUNCTIONS. 53 

Hence, on letting Ax approach zero, 

dy = F(x)<l>'(x)-<Kx)F'(x) m 

dx [^(*)] 2 

That is : If one function be divided by another, then the derivative 
of the fraction thus formed is equal to the product of the denomi- 
nator by the derivative of the numerator minus the product of the 
numerator by the derivative of the denominator, all divided by 
the square of the denominator. 

If the functions be denoted by u and v ; that is, if 

u 
y = v' 
then (1) has the form 

v du_ u d*L 
dy _ doc doc (2) 

doc ~ v% 

The differential of the quotient of two functions. If y = -, then 



from (2) and the definition in Art. 27, 



v 



v — dx — u — dx /Q s 

dx dx \y) 

3 ~ v 2 

But, by Art. 27, — dx = du and — dx = dv. Hence (3) may 
be written 

dy = vdu ~ 2 udv . (4) 

Note. The derivative (1), or (2), can also be obtained by means of Art. 

32. For if y = -, then vy = u. Whence v ^- + y~ = —- From this 
v dx dx dx 

_y = — Vl — V. Jt % which reduces to the form in (2) on substituting - for y. 
dx v dx v dx v 

Ex. 1. Find the derivatives and the differentials of 
a 3 x 2 + 7 x - 11 



Sx 2 -7 x + 2' z 3 + 8' 2x 2 -9z + 3 

Ex. 2. Calculate the differentials of the functions in Ex. 1 when x = 2 
and dx = .1. 



54 DIFFERENTIAL CALCULUS. [Ch. IV. 

34. The derivative of a function of a function. 

Suppose that y = 4>(u), 

and that u = F(x), 

and that the derivative of y with respect to x is required. (Here 
4>(u) and F(x) are differentiate functions.) The method which 
naturally comes first to mind, is to substitute F(x) for u in the 
first equation, thus getting y = <$>\_F(x)\ and then to proceed 
according to preceding articles. This method, however, is often 
more tedious and difficult than the one now to be shown. 

Let x receive an increment Ax; accordingly, u receives an incre- 
ment Au, and y receives an increment Ay. Then 



y + Ay = <f>(u + Au). 






.'. Ay = <j>(u + Aw) - 


- <j>(u). 




Ay <j>(u + Ait) - 


-Cf>(u) 




Ax Ax 






cj>(u + Au) - 


-cf>(u) 


Au 


Au 




Ax 



Assume A?j =£ when Ax =£ 0. When Ax approaches zero Au 
approaches zero, and this relation becomes 

dy d r »/ m du 
dx die )A dx 

. dy = dydu m (±) 

due du doc v ' 

Note. It should be clearly understood that the first member of (1) does 
not come, and cannot come, from the second member by cancellation of the 
dw's. Cancellation is not involved at all. 

Result (1), which may be expressed more emphatically (Art. 23), 

is an important one and has frequent applications. It may be thus stated : 
the derivative of a function with respect to a variable is equal to the product 
of the derivative of the function with respect to a second function and the 
derivative of the second function with respect to the first named variable. 
(Here all the functions concerned are supposed to be diff erentiable. ) 



34,35.] DIFFERENTIATION OF FUNCTIONS. 55 

From (1) and (2) it results that 

A_ {) dy 

jL (v >> = <te ie <*£ = <**. (3) 

du KJJ A-( U -) ' ' du du 
dx J dx 

Relations (1) and (2), Note 1, Art. 26, are special applications of (1) [or 
(2) and (3)]. The showing of this is left as an exercise for the student. 

Ex. 1. Explain why the du's in (1) may not be cancelled. 

Ex. 2. Eind ^, given that y = u s and u = x 2 + 1. 
dx 

Here ^M. = Su 2 ,—=2x. . \ ^ = 6 u 2 x = 6 x(x 2 + l) 2 . 
du dx dx 

Ex. 3. Find -^- when y = 3 u 2 and u = $c 4 — 3 x -f 7. Verify the result 
dx 
by the substitution method referred to at the beginning of the article. 

Ex. 4. Find — when z = 2 v 2 - 3 v + 1 and v = 6 1 2 + 1. Verify the 
dt 
result by the substitution method. 

Ex. 5. Show that a function of a function is represented by a curve in 
space. (See Echols, Calculus, Appendix, Note 2.) 

35. The derivative of one variable with respect to another when 
both are functions of a third variable. 

Let x = F{t) and y = <f>(t). 

Now —^ = ^1-5 Now At, Ax, and Ay reach the limit zero 

Ax At At 

together. (Assume that Ax ^0 when Ay^O.) 
Hence, on letting At approach zero, 

dy 

dy=—. (i) 

dx dx 

dt 

This result may also be derived as a special case of result (3), 
Art. 34. This is left as an exercise for the student. 

Ex. 1. Find % when y = 3 t 2 - 7 t + 1, and x = 2 t 3 - 13 1 2 + 11 1. 
dx 

Here ^M=6t-7, ^ = 6t 2 - 26t + 11. .-. ^ = 6t ~ 7 

dt dt dx 6t 2 -26t + 11 

Ex. 2. Find ^ when x = 2 t 2 + 17 t - 1 and y = 3 1* - 8 t 2 + 9. 
Ex. 3. Find — when u = 7 x 4 - 3 and v = 3x 2 + 14x - 4. 



56 DIFFERENTIAL CALCULUS. [Ch. IV. 

36. Differentiation of inverse functions. If y is a function of x, 
then x is a function of y ; the second function is said to be the in- 
verse function of the first. This is expressed by the following nota- 
tion: If y=f(x), then x=f~ 1 (y). Assume that the function fix) 
and its inverse f~ x (y) are continuous and also differentiable. 

For cases in which Ax =£ when Ay=^0 it follows from the 

equation — • — = 1, since Ax and Ay approach zero together, 
Ax Ay 

dx dy 

Hence, in such cases, doc ~ dec' 

dy 

DIFFERENTIATION OF PARTICULAR FUNCTIONS. 

In the following articles u denotes a continuous function of oc 9 

and differentiation is made with respect to x. The letters a, n 9 •••, 
may denote constants. 

N.B. It is advisable for the student to try to obtain the derivatives before 
having recourse to the book for help. 

A. Algebraic Functions. 

37. Differentiation of u n . 

(a) For n, a positive integer. 
Put y = u n ) 

i.e. y = uuu ••• to n factors. 

... 4/ = u n-l du + u n-l du + m m m tQ n termg . Arf . S2) 

dx dx dx 

n -\du 
= nu n — • 
dx 

d d 

In particular, — (x) — 1, and — (x n ) = nx n ~\ 
dx dx 

Ex. 1. Give the derivatives with respect to x of 

w 2 , 3u±, 7w 9 , x 8 , Zx\ 7 a 12 , 9 x* - 17 a 2 + 10 x + 40. 



36, 37.] DIFFERENTIATION OF FUNCTIONS. 51 

Ex. 2. Find the ^-derivative of (2x + 7) 18 . 

On denoting this function by y, and putting u for 2 x + 7, y = u 18 . Hence 

dx dx 

Now— = 2; hence ^ = 36 u 17 = 36 (2x + 7) 17 . 
dx dx 

The substitution u for 2 x + 7 need not be explicitly made. For, if 
y= (2x + 7)18, 

then ^ = 18 (2 x + 7) 17 — (2 x + 7) (Art. 34) 

dx dx 

= 36 (2 x + 7) 17 . 
Ex. 3. Differentiate 

(5 x 2 - 10) 2 *, (3 x 4 + 2)io, (4 X 2 + 5) 8 (3 x* - 2 x + 7) 5 . 
(6) For n, a negative integer. Let n = — m, and put ?/ = u n . 
Then V = ir m = — 

u™ . A(i)_i . A( M ») 

,\ * = — = — (Art. 33) 

— mu m 1 — 

dx . . . . .dw 

= 5- = (— m) w (- wt )- 1 — 

it 2w K } dx 

= nu n L — ■• 
dx 

Ex. 4. Differentiate with respect to x, 

w- 2 , u- 7 , w-n, x-7, 3x- 5 , 17x-i<\ (x 2 -3)-S (3x* + 7)-6, 

"3s 5 -7x 3 + 2-- + — _ J_. 

x x 2 9 x 3 

P 
(c) For n, a rational fraction. Let n = — , in which p and <? 

are integers. 
p 
Put y=u q ; then ?/ 3 = w p . 

On differentiating, qy*' 1 ^- = pu p ~ 1 —' 

dx dx 

. % _ ff ?/y ~ 1 du _p u 1 "- 1 du _ p *~ l du__ ?n -i^w. 
dx q y 1 ' 1 dx q %-d dx q dx dx 



58 DIFFERENTIAL CALCULUS. [Ch. IV. 

Ex. 5. Find the ^-derivatives of 

f— 1 _3 7 !- 7 , . . 

vw (i.e. w 2 ), u 4 , u 5 , vx, x 2 , vx 5 , V3x 2 — 5, 

\/2x 2 + 7x-3, V2x+7, (3x-7)~^, 3x 2 - 7x^ + -^ + ~~-^- 

x~z S 7x% 

(d) For n, an incommensurable number. In this case it is also 

true that — (u n )=nu n - 1 —. This is proved in Art. 39 (b). 
dx dx 

Hence, for all constant values of n, 

£ {u ^ =nU n-^ (1) 

In particular, if u = x, — (x n ) = nx n ~ l . 
dx 

Ex. 6. Find the x-derivatives of 

v/\ x v ~\ 5/', (2x4-5)^, (3 x 2 + 7 x - 4)^. 

Ex. 7. Write three functions which have x 3 for a derivative. 

Ex. 8. Do as in Ex. 7 for the functions 

x 5 , I, Vx, Vx3, vx, 6x 4 -- - ^L. 

x 2 x 2 Vx 

Ex. 9. Show that £fte general form ichich includes all the functions that 

x n ^ 

have x n for the derivative, is h c, in which c is an arbitrary constant. 

n + 1 

Note 1. The result (1) and the general results, Arts. 29-36, suffice for 
the differentiation of any algebraic function. 

Note 2. Case (a) can also be treated as follows : Put y = u n , and let x 
receive an increment Ax ; then u and y receive increments Au and Ay 
respectively. Then y + Ay = (u + Au) n . On expanding the second member 

by the binomial theorem, then calculating Ay and then — ^, and finally letting 

Ax 
Ax approach zero, the result will be obtained. 

Note 3. It is well to remember that — (x) =1 and — (Vx) = • 

dx dx 2 Vx 

Ex. 10. Do the operations indicated in Note 2. 



xvx 2 4- 7 
Ex. 11. Differentiate ^— Find the value of the derivative when 

x = 2. V^+2 

Put , = «(* + 7)*. > 

(x 2 + 2) 3 



37.] DIFFERENTIATION OF FUNCTIONS. 59 

(x 2 + 2)^— [>(x 2 + 7 )ij _ X ( X 2 + 7)i^_( x 2 + 2)3 
Then ft = ^ - ^ 

^ x (x 2 + 2)3 

On performing the differentiations indicated in the second member, and 
reducing, it is found that 

dy = 4 x 4 + 19 x 2 + 42 

dx 3 (x 2 + 7)*(x 2 + 2)* 



Hence, when 



- 1.68, approximately. 



dx 

Ex. 12. Differentiate the following functions with respect to x : 

(2x-5)(x 2 + llx-3), ax»+-, ^-±-^, 2Lz_* VIT^ 2 , £ + 5 v^ -7 x 5 , 

x" 1 - x 2 a + x x 4 



Vl + X 2 X X 3 J 1 



x \/a — bx 2 n _ x i\i 1 — x 



(a + x) Va — x. 

Ex. 13. Find -^ when x 2 y 3 + 2x + 3y = 5. Here y is an implicit function 
dx 

of x. On differentiation of both members with respect to x, 

x 2 ^(y*)+y s -f(x 2 ) +2 + 3-^ = 0; . 
dx dx dx 

i.e. 3 *V— + 2 x?/ 3 + 2 + 3 ft = 0. 



dx dx 



Erom this 



dy _ 2 (1 + xy 3 ) f 

dx~ 3(l + x 2 2/ 2 )' 



Ex. 14. (a) Find -^ when x and y are connected by the following rela- 
dx 

tions : y s + x 3 — 3 ax?/ = ; x 4 + 2 ax 2 y — ay 3 = 0; 7 x 2 y 2 + 2 x?/ 3 — 3 x B y + 4 x 2 

- 8 ?/ 2 = 5 ; (a + ?y) 2 (6 2 - y 2 ) + (x + a) 2 ?/ 2 = ; x 2 + y 2 = a 2 ; a 2 y 2 + 6 2 x 2 = 

a 2 b 2 . In the last case also obtain -^ directly in terms of x. 

dx 

(&) In the ellipse 3x 2 + 4y 2 = 7, find the slope at the points (1, 1), 

(1, -1), (-1,1), (-1, -1). 

N.B. The following examples should all be worked by the beginner. 
They will serve to test and strengthen his grasp of the fundamental prin- 
ciples of the subject, and will give him exercise in making practical applica- 
tions of his knowledge. For those who may not succeed in solving them 



60 DIFFERENTIAL CALCULUS. [Ch. IV. 

after a good endeavour, two examples are worked in the note at the end of 
the set. 

Ex. 15. A ladder 24 feet long is leaning against a vertical wall. The foot 
of the ladder is moved away from the wall, along the horizontal surface of 
the ground and in a direction at right angles to the wall, at a uniform rate 
of 1 foot per second. Find the rate at which the top of the ladder is descend- 
ing on the wall when the foot is 12 feet from the wall. 

Ex. 16. Show that when the top of the ladder is 1 foot from the ground, 
the top is moving 575 times as fast as when the foot of the ladder is 1 foot 
from the wall. 

Ex. 17. Find a curve whose slope at any point (#, y) is 2x. Find a 
general equation that will include the equations of all such curves. Find 
the particular curve which passes through the point (1, 2). 

Ex. 18. A man standing on a wharf is drawing in the painter of a boat at 
the rate of 4 feet a second. If his hands are 6 feet above the bow of the boat, 
how fast is the boat moving when it is 8 feet from the wharf ? 

Ex. 19. A man 6 feet high walks away at the rate of 4 miles an hour from 
a lamp post 10 feet high. At what rate is the end of his shadow increasing 
its distance , from the post ? At what rate is his shadow lengthening ? 

Ex. 20. A tangent to the parabola y 2 = 16 & intersects the x-axis at 45°. 
Find the point of contact. 

Ex. 21. A ship is 75 miles due east of a second ship. The first sails west 
at the rate of 9 miles an hour, the second south at the rate of 12 miles an 
hour. How long will they continue to approach each other ? What is the 
nearest distance they can get to each other ? 

Ex. 22. A vessel is anchored in 10 fathoms of water, and the cable passes 
over a sheave in the bowsprit which is 12 feet above the water. If the cable 
is hauled in at the rate of a foot a second, how fast is the vessel moving 
through the water when there are 20 fathoms of cable out ? 

Ex. 23. Sketch the curves y 2 = 4 x and x 2 = 4 y, and find the angles at 
which they intersect. (If 6 denotes the angle between lines whose slopes 
are m and n, tan 6 = (m — ri) -s-(l + mn) ; see analytic geometry and plane 
trigonometry.) 

Ex. 24. Sketch the curves y 2 = 8 x and x 2 = 8 y, 
and find the angles at which they intersect. 

Note. Examples worked. Ex. 15. Let FT be 

the ladder in one of the positions which it takes during 
the motion, and let FH be the horizontal projection of 
FT. Let FH=x, and HT=y. Then 

x 2 + y 2 = 576. (1) Fig. 10. 




38.] DIFFERENTIATION OF FUNCTIONS. 61 

dx 

Now x and y are varying with the time ; the time-rate — is given, and 

du ^ 

the time-rate -^ is required. Differentiation of both members of (1) with 

dt 
respect to the time give 

2x^+2^ = 0; 
dt dt 

whence dy = _xte (2) 

dt y dt 

dx 
In this case, — =1 foot per second, x = 12 feet, and, accordingly, 
dt 



V = V24 2 - 12* feet = 12 V3 feet. 

/. -^ = — • 1 foot per second = — .577 feet per second. 

& 12 V3 

The negative sign indicates that y decreases as x increases. It should be 
noticed that the result (2) is general, and that all particular solutions can 

dx 
be derived from it by substituting in it the particular values of x, y, and — 

dt 

Ex. 17. Find a curve whose slope at any point (x, y) is 2x. Find a 

general equation that will include the equations of all such curves ; and find 

the particular curve which passes through the point (1, 2). 

Here ^ = 2 x. 

dx 

Hence y = x 2 + c, (1) 

in which c denotes any arbitrary constant. This is the general equation of 
all the curves having the slope 2 x. .-. y = x 2 + 7 is one of the curves ; 
y = x 2 — 5 is another. If the point (1, 2) is on one of the curves (1), then 
2 = 1 + c ; whence c = 1, and, accordingly, y = x 2 + 1 is the particular curve 
passing through (1, 2). As in Ex. 15 it is easier to find first the general solu- 
tion of the problem in question, and therefrom to obtain any particular 
solution that may be required. Figure 9 shows some of these curves. 

B. Logarithmic and Exponential Functions. 

38. Note. To find lim m =x [ 1 H — ) • This limit is required in what 
follows. " V ml 

(a) For m, a positive integer. By the binomial theorem, 

V m) m 1-2 m 2 L2-3 m» w 

This can be put in the form 

(i+Ay=i + i + A_H2 + J — md\ — md + .... (2) 



62 DIFFERENTIAL CALCULUS. [Ch. IV. 

On letting m approach infinity, and taking the limits, this becomes * 

V m) 2 ! 3 ! 

= 2.718281829.-.. (3) 

This constant number is always denoted by the symbol e. 

(b) The result (3) is true for all infinitely great numbers, positive and 
negative, integral, fractional, and incommensurable. For the proof of (3) 
for all kinds of numbers, see Chrystal, Algebra (ed. 1889), Part II., Chap. 
XXV., §13, Chap. XXVIII., §§ 1-3; McMahon and Snyder, Biff. CaL, 
Art. 30, and Appendix, Note B ; Gibson, Calculus, § 48. 

Note on e. The transcendental number e frequently presents itself in 
investigations in algebra (for instance, as the base of the natural logarithms, 
and in the theory of probability), in geometry, and in mechanics. The num- 
bers e and ir are perhaps the two most important numbers in mathematics. 
They are closely allied, being connected by the very remarkable relation 
e iir = — l,t which was discovered by Euler. See references above, and KleiD, 
Famous Problems (referred to in footnote, Art. 8), pages 55-67. 

39. Differentiation of log a u. 

Put y = \og a ii, 

and let x receive an increment Ax ; then u and y consequently 
receive increments Au and Ay respectively. 

Then y -f Ay = \og a (u + Au). 

.-. Ay = \og a (u-j-Au) — log a u 



K^) =l0 K 1+ v) 



A?/ i /., , Au\ 1 
.-. _£ = log.[l + — ).— . 
Ax \ u.J Ax 

On introducing Au in the second member, 

u Au 

u 

A?/ 1 u i /-, . Au\ Au 1 -. A, . AuX±u Au 
_^ = _._log a (l + — . — = -log a 1 + — •— • 
Ax u Au V u Ax u V u I Ax 



* This conclusion is properly reached only after a more rigorous investiga- 
tion than is here attempted. (See Arts. 167-171.) 
t See Art. 153. 



39.] DIFFERENTIATION OF FUNCTIONS. 63 

From this, on letting Ax approach zero and remembering that Au 
and Ay approach, zero with Ax, it follows by Arts. 22, 23, 38, that 

dy 1 i du 

■— = — ' log a e ; 

dx u dx 

i.e. 4~ (1<>S« *«) = -• log« e . ^. 

If w = x, then — — (loga m) = — • loga e. 

to X 

If a = e, then #- (log u) = ±^, 

doc udoc 

I£u=x, and a=e. then — — (log x) = — . 
to a? 

Note. When e is the base it is usual not to indicate it in writing the 
logarithm. 

Ex. 1. Find the derivatives of log a (3 x 2 + 4 x - 7), log (3 x 2 + 4 x - 7), 
logio (3 x 2 + 4 x — 7). Find the values of these derivatives when x = 3. 

Ex. 2. Find the values of the derivatives of log Vx 3 + 10, logio Vx 3 + 10, 
when x = 2. 

Ex.3. Differentiate the following: log^^, log\ft-±-^, log 1 + _ , 
log (x + Vx 2 + a 2 ) , log (log x) , x log x. 1 + x 1_:c 1-Vx 

Ex. 4. Find anti-derivatives of 2 * + 3 3 x 2 - 7 



x 2 + 3 x + 5 x 3 - 7 x - 1 2 x 
(a) Logarithmic differentiation. If 

?/ = uvw, (1) 

then log y = log w + log v + log w. 

On differentiation, 1^ = 1 *L + 1 * .+ 1 *° 

2/ dx w dx v dx w da? 



whence -^ = Mmo 

dx 



1 dw 1 dv 1 dwfl /o\ 

u dx v dx w dx J 



This result can easily be reduced to the form obtained in 
Art. 32. The same method can be used in the case of any finite 
number of factors. This method of obtaining result (2) is called 



64 



DIFFERENTIAL CALCULUS. 



[Ch. IV. 



the method of logarithmic differentiation. It is frequently more 
expeditious than that given in Arts. 32, 33, especially when 
several factors are involved. 



Ex. 5. Find ^ when 
dx 



Here, 

On differentiation, 



y _ %(x 2 + 7)^ (See Ex> 11} Art 37) 

(a 2 + 2)* 
logy = logz + ilog (a; 2 + 7) - *log (a 2 + 2). 

!$/__! I # 2x 



y dx x x 2 + 7 3 (z 2 + 2) 
From this, on transposing, combining, and reducing, 

4 x 4 + 19 x 2 + 42 






3 (x 2 + 7)*(x 2 + 2) 



Ex. 6. Differentiate, with respect to x, the following functions 



(a) 



(«+2)» , 5 . (g-l)(g-2) . f v 



(4x-7)3(3x + o)^ 



(» + !)(» + 2) 



\/2x + 5\ / 7x-5 | 
v^(* + 3)2 



(6) Differentiation of an incommensurable (constant) power of a 
function. This paragraph is supplementary to Art. 37 (d). 

Let y - yjn 9 

in which n is any constant, commensurable or incommensurable. 

log y = n log u. 



Then 
From this 

and hence 



Idy _ndu m 
ydx udx' 

dx u dx dx 



Note. This deri- 
vation assumes that 

-^ exists. 
dx 



40. Differentiation of a™. 



Put 
Then 

On differentiation, 



i.e. 



y = a\ 
log y = u log a. 

Idy i d^ 

ydx dx 

dy i c?w 

^-(a**)=a«.loga-^ 



(See Note above.) 



40.] DIFFERENTIATION OF FUNCTIONS. 65 



If u = x, then 




If a = e, then 


£<«?■>-*£ 


If u = x, and a = 


= e, then 




£(-) = -! 



that is, the derivative of e x is itself e x . 



Note 1. On the derivation of results in Arts. 39, 40. The derivative 
of log a u was deduced by the general and fundamental method, and has 
been used in rinding the derivative of a u . The latter derivative can be 
found, however, by the fundamental method, independently of the deriva- 
tive of log a u. Moreover, the derivative of log a u can be obtained by means 
of the derivative of a u . These various methods of finding the derivative 
of a u and log a u are all employed by writers on the calculus. For examples 
see Todhunter, Diff. Cal., Arts. 49, 50; Gibson, Calculus, §65, where both 
these derivatives are obtained independently of each other ; Williamson, 
Diff. Cal, Arts. 29, 30; McMahon and Snyder, Diff. Cal., Arts. 30, 31, 
where the derivative of the logarithmic function is first obtained and the 
derivative of the exponential function is deduced therefrom ; and Lamb, 
Calculus, Arts. 35 (Ex. 5), 42, where the derivative of the exponential 
function is obtained first and the derivative of the logarithmic function 
is deduced therefrom. (See also Echols, Calculus, Art. 33 and foot-note.) 

Note 2. On the expansion of e x in a seines see Hall and Knight, Higher 
Algebra, Art. 220 ; Chrystal, Algebra, Vol. II., Chap. XXVIII., §§ 4, 5; and 
other texts. (This expansion is derived by the calculus in Art. 178, Ex. 7.) 

Ex. Assuming the expansion for e x , show that the derivative of e x is 
itself e x . 

Note 3. The compound interest law. The function e x "is the only 
[mathematical] function known to us whose rate of increase is proportional 
to itself ; but there are a great many phenomena in nature which have this 
property. Lord Kelvin's way of putting it is that ' they follow the compound 
interest law.' " (See Hall and Knight, Higher Algebra, Art. 234, and, in 
particular, Perry, Calculus, Art. 97 and Art. 98, Exs. 4, 2.) 

Ex. 1. Differentiate, with respect to x, e x \ 10 x , 10 3 * 2 , e v *. 

Ex. 2. Find the ^-derivatives of e 2 *, 10 f2 , / +3 , 10 2 ' +7 . 

Ex. 3. Find the ^-derivatives of the following : 

n T, P x P~ x fi x2 

e x x m , a x , — — , xe~ x , — , — . 

e x — 1 e x + e~ x x 

Ex. 4. Find anti-derivatives of e" ,x , xe x2 , 2 e 3x+1 . 



66 DIFFERENTIAL CALCULUS. [Ch. IV. 

41. Differentiation of u v , in which u and v are both functions 
of oc. 

Put y = u\ (1) 

Then log y = v log u. 

On differentiation, 1 & = * *• + log u ■ *• 

y dx u dx dx 

dy_ fv du i dv\ t 

dx \w da? " cfay ' 

"■ £<-».-»g£+*r.£). (2) 

Note 1. It is better not to memorize result (2), but merely to note the 
fact that the function in (1) is easily treated by the method of logarithmic 
differentiation. 

Note 2. The beginner needs to guard against confusing the derivatives 
of the functions w n , a u , and u v . 

cJv 
Ex. 1. Find -—- when y = x x . 

Here log y = x log x. 

1 civ X 
On differentiation, - ~ = - + log x ; 

y dx x 

whence -=- = x x (l + logic). 

Ex. 2. Eind the ^-derivatives of 

(3z + 7)* 2 , (3z + 7) 2x , {(3a; + 7) x }*, ^x, x* n , e>\ (*)l log*. 

\x/ a x 

C. Trigonometric Functions. 

42. Differentiation of sinu. 

Put ?/ = sin u. 

Then y -\- Ay = sin (w + Aw). 

.*. Ay = sin (w -f Au) — sin w 

= 2 cos [ w + -£■ j sin -^. (Trigonometry) 



41, 42. ] DIFFERENTIATION OF FUNCTIONS. 67 

Ay ( . Au\ • Au 1 
.'. — - = 2 cos [ u + — — sin — • — • 
Ax \ 2 J 2 Ax 

• Ait 
sin — - 

/ , Au\ 2 Aw 

== cos [ u + 



2 7 Aw Ax 



2 
Let A# = ; then also Au = 0, and 



Au 



sin 

lim As -o -^ = lim AM -o cos (u + — ^ ] • lim Att=y) — • lim A ^ ~ ; 

Aa; V 2 J Au Ax 



dy ., du 

-JL = cos m • 1 ; 

c?a? dx 



2 



i.e. 4- (sin u ) = cos «* f^. (1) 

ddo dx 

In particular, if u = x, 

4- (sin x) = cos a?. (2) 

dx 

That is, the rate of change of the sine of an angle with respect 
to the angle is equal to the cosine of the angle. 

Note 1. Result (2) can also be obtained by geometry. (Ex. Show this.) 
See Williamson, Diff. Cal., Art. 28, and other texts. 

Note 2. Result (2) shows that as the angle x increases from to — the 
rate of increase of the sine is positive, since cos x is then positive. As x 
increases from - to ir the rate is negative (i.e. the sine decreases), since 

2 q _ 

cos x is then negative. The rate is negative when x increases from ir to '——, 

and the rate is positive when x increases from — - to 2 ir. This agrees with 

what is shown in elementary trigonometry, and it is also apparent on a 
glance at the curve y = sin x. 

Note 3. Result (2) also shows that if the angle increases at a uniform 
rate, the sine increases the faster the nearer the angle is to zero, and 
increases more slowly as the angle approaches 90°. This is also apparent 
from an inspection of a table of natural sines, or from a glance at the curve 
y = sin x. 

Note 4. The derivative of sin if has been found by the general and 
fundamental method of differentiation. It is not necessary to use this 



68 DIFFERENTIAL CALCULUS. [Cn. IV. 

method in finding the derivatives of the remaining trigonometric and anti- 
trigonometric functions, for these derivatives can be deduced from that of 
the sine. 

Ex. 1. Find the ^-derivatives of sin 2 w, sin 3 u, sin \ u> sin § w, sin \ 7 - u. 
Ex. 2. Eind the x-derivatives of sin2x, sin3x, sin|x, sin3x 2 , sin 2 3x, 
sin4x 5 , sin 5 4x. 

Ex. 3. Eind the derivatives with respect to t of sin 5 1, sin \ t 2 . 

Ex. 4. Eind the ^-derivatives of sm 2 x , xsin2x, x 2 sin( x + -V 

sin 3 x V 4 / 

Ex. 5. At what angles does the curve y = sinx cross the x-axis ? 

Ex. 6. At what points on the curve y = sin x is the tangent inclined 30° 
to the x-axis. 

Ex. 7. Draw the curve y = sin 2 x. At what angles does it cross the 
x-axis ? 

Ex. 8. Draw the curve y = sin x + cos x. Where does it cross the x-axis ? 
At what angles does it cross the x-axis ? Where is it parallel to the x-axis ? 

Ex. 9. Eind the x-derivatives of the following: sin nx, sinx n , sin"x, 
sin(l+x 2 ), sin(wx + a), sin(a -j- 6x n ), sin 3 4x, sina; , sin(logx), log(sinx), 
sin(e*) • logx. 

Ex. 10. (a) Find anti-derivatives of 

cosx, cos3x, cos(2x + 5), xcos(x 2 — 1). 

(6) Find anti-differentials of cos2x$x, cos(3x — 7)dx, x 2 cosx B dx. 

Ex. 11. Calculate d(sinx) when x = 46° and dx = 20', and compare the 
result with sin 46° 20' — sin 46°. (Radian measure must be used in the 
computation.) 

Ex. 12. Compare d(sinx) when x = 20° and dx = 30', with 

sin 20° 30' - sin 20°. 

43. Differentiation of cos u. 



Put 


y = cos u. 




Then 


2/ = sin^|-A 






dx \2 Jdx\2 


-„) 




dx 





[Art. 42, Eq. (1)] 



i.e. -^-(cosw) = -sinw^. Yl^ 

cZx doc w 



43,44.] DIFFERENTIATION OF FUNCTIONS. 69 

In particular, if u = x, 

— - (cos x) = - sin x. (2) 

ax v ' 

Ex. 1. Obtain derivative (1) by the fundamental method. 

Ex. 2. Show that result (2) agrees in a general way with what is shown 
in trigonometry about the behaviour of the cosine as the angle changes from 
0° to 360°. Also inspect the curve y = cos x. 

Ex. 3. Find where the curve y = cosx is parallel to the x-axis, and where 
its slope is tan 25°. 

Ex. 4. Show that the tangents of the curve y = cosx cannot cross the 
x-axis at an angle between + 45° and + 135°. 

Ex. 5. Find the slope of the tangent to the ellipse x = a cos 0, y = b sin 6. 
(See Art. 35.) 

Ex. 6. Find the slope of the tangent to the cycloid x — a (6 — sin 6), 
y = a{\ — cos 8). What angle does this tangent make with the x-axis when 

a = 5, and = - ? 
3 

Ex.7. Find the x derivatives of the following: cos(2# + 5), cos 3 5 as, 

x 2 cos x, — = , cos mx cos nx, xe cos x , e ax cos rax. 

> + cos x 

Ex.8. Find anti-differentials of sinxdx, sin^xdx, sin(3x — 2)dx, 
xsin(x 2 + 4)c?x. 

Ex. 9. Calculate d cos x when x = 57° and dx = 30', and compare the 
result with cos 57° 30' - cos 57°. 

44. Differentiation of tan u. 

Put y = tan u. 

Then sin«. 

COS u 

cos u — (sin u) — sin u — (cos u) 
dy_ dx y J dx K J 



dx 



_ (cos 2 u + sin 2 u) du 
cos 2 u dx 

1 du o du 



— oen 2 



sec' u — ; 
cos 2 u dx dx 

i.e. 4~ ( tan u ) = sec2 u %r* C 1 ) 

dx dx K ' 



70 DIFFERENTIAL CALCULUS, [Ch. IV, 

If u = x, then -p- (tan a?) = sec 2 x. (2) 

Ex. 1. Show the agreement of result (2) with the facts of elementary- 
trigonometry, and with the curve y = tan x. 

Ex. 2. Show that the tangents of the curve y = tan x cross the x-axis at 
angles varying from + 45° to + 90°. 

Ex. 3. State the x-derivatives of tan 2 u, tan 3 u, tan mu, tan na 2 , tan 2 x, 
tan i x, tan w»x, tan 3 x 2 , tan 4 x 3 , tan mx n , tan 2 3 x, tan 3 4 x, tan n mx, 

tan 2 (fx + 3), log tan-. 

2 

Ex. 4. Find anti-differentials of sec 2 xdx, sec 2 2 x cfcc, sec 2 (3 x + d)dx. 

Ex. 5. Compute d tan x when x = 20°, dx = 20', and compare the result 
with tan 20° 20' - tan 20°. 

Ex. 6. When is the differential of tan x infinitely great ? 

45. Differentiation of cot u. 

Either, substitute -, for cot u, and proceed as in Art. 44 ; 

sin u 

or, substitute tan (90°— u) for cot u, and proceed as in Art. 43; 

or, substitute for cot u, and differentiate. It will be 

found that tan u 

-^- (cot u) = - cosec 2 u§±- (1) 

dx dx 

Ifu = x, ^~ (cot as) = - cosec 2 as. (2) 

dx v y 

Ex. Show the general agreement of result (2) with the facts of ele- 
mentary trigonometry, and with the curve y = cot x. 

46. Differentiation of sec u. 

1 



Put y = sec u = 

cosu 

rpi dy _ sin u du _ 1 sin u du t 

dx cos 2 ^ cfce cos u cos w da; ' 

i.e. — (sec i«) = sec u tan w — . (1) 

dx dx K ' 

If u = x, ■— (sec x) = sec x tan a?. (2) 

ax 



45-49.] DIFFERENTIATION OF FUNCTIONS. 71 

47. Differentiation of esc u. 



Put 


y = esc u = Then -%■ = - 

sm u dx 


cos u du 
sin 2 w cto 




That is, 


d / x , du 

— (esc u) = — esc u cot u 

dx dx 




(i) 


If u = X, 


— (esc x)= — esc x cot X. 




(2) 



Note. Or put y = esc u = sec ( — — u J , and proceed as in Art. 43. 

48. Differentiation of vers u. Put y = vers u = 1 — cos w. Then, 
on differentiation, ^ ^ 

— (vers u) = sin w — • 
dx dx 

In particular, if u = x, 

— — (vers oc) = sin x, 
doc 

Ex. 1. Find the x-derivatives of cot (2 x + 3), sec (| x + 3), esc (3 x — 7), 
vers (5 x + 2), sec n x. 

Ex. 2. Find the ^-derivatives of cot 2 (3 * + 1), sec 3 (J « - 1), esc 2 f (« + 5), 
cot(9f 2 ), sec (7 «-2) 2 . 

Ex. 3. Show that D log (tan x -f sec x) = D log tan (i 7r + i x) = sec x. 

Z). Inverse Trigonometric Functions.* 

49. Differentiation of sin -1 */. 

Put y = sin -1 u. 

Then sin y = u. 

On differentiation, cos ?/ ^ = — • 



dy _ 1 du _ 1 (fot. 

cZ# cos 2/ cto Vl — sin 2 « ^' 

?.e. # (sin-i u) - * ^. (1) 

da? Vl - u* doc x 7 

If M = a- A (sin - 1 x) = 1 . (2) 

<**> Vl-oc* 

* See Murray, Plane Trigonometry, Arts. 17, 88. 



72 



DIFFERENTIAL CALCULUS. 



Note 1. On the ambiguity of the derivative of 
sin-l as. The result in (2) is ambiguous, since the sign of 
the radical may be positive or negative. This ambiguity 
is apparent on looking at the curve y = sin -1 x, Fig. 11. 

Draw the ordinate ABCDE at x = X\. The tangents 
at B and D make acute angles with the x-axis, and the 
tangents at C and E make obtuse angles with the x-axis. 

Hence, at B and D -^ is positive ; and at C and E -^ is 
dx dx 

negative. That is, at B and D — (sin -1 x) = — + ; 

dx Vl — i-- 2 



and at C and E — (sin -1 x) = 

dx K VT~ 

d 



Xf 

Thus the sign 




xr 



of — (sin -1 x) depends upon the particular value taken of the infinite number 
of values of y which satisfy the equation y = sin -1 x. 

Note 2. If it is understood that there be taken the least positive value of 
y satisfying the equation y = sin -1 Xi (in which x\ is positive), then the sign 
of the derivative is positive. Similar considerations are necessary in (1). 

Ex. 1. Show by the graph in Fig. 14, or otherwise, that when x = 1, 

— (sin -1 x) = + oo, and that when x = — 1, — (sin -1 x) is — co. 
dx dx 



Ex. 2. Find the ^-derivatives of 



sin -1 x n , sin -1 



a + 1 

V2 



sin - 



2x 



,2' 



snr 



2x 



sin- 1 VI - x 2 , Vl 



1 + x 2 ' vT-a? 

x 2 • sin -1 x — x, sin -1 Vsin x. 



Ex. 3. Show that a tangent to the curve y = sin -1 x cannot cross the 
x-axis at an angle between — 45° and + 45°. 

1 2x x 2 



Ex. 4. Find anti-derivatives of 



Vl -x 2 Vl -x 4 Vl - x 6 



50. Differentiation of cos -1 u . 

Put y = cos -1 u. 

Then cos y — u. 

On differentiation, - sin y ^ = — • 

dx dx 



.dy 
' dx 



1 du 



du 



sin y dx ^l — cos 2 y dx ' 



50, 51.] DIFFERENTIATION OF FUNCTIONS. 73 

i.e. 4- (cos -1 »> = - 1 ^. 

<fc» Vl - u* dx 

If u = x, A(cos-i) - 



dx ' Vl - a;2 

Ex. 1. Explain the ambiguity of sign in the derivative of cos -1 a by 
means of the curve y = cos -1 x. Show that if there be taken the least 
positive value of y satisfying y = cos -1 x, in which x is positive, the sign 
of the derivative is negative. 

Ex. 2. Determine the angles at which the tangents touching the curve 

ii — cos -1 x where x — — , cross the x-axis. 

V2 

rfln 1 1 /v»2 fl rf 

Ex. 3. Eind the ^-derivatives of cos" 1 -, cos" 1 - — — , a cos -1 - — -• 

X 2n _|_ i' 1 + x 2 a 

51. Differentiation of tan -1 */. 

Put y = tan -1 u. 

Then tan y = u. 

On differentiation, sec 2 y -^ = — • 

cfa eta 

^— 1 du _ 1 cfoi . 
do; sec 2 ?/ cto 1 + tan 2 y dx ' 

i.e. ^. (t an-^)=-l-^. 

dx 1 + i*2 ^a? 

In particular, if ?< = x, 

-f- (tan-i a?) = 



dx l + x% 

Note. The derivative of tan -1 x is always positive. This is also evident 
on a glance at the curve y = tan -1 x. 

Ex. 1. Eind the ce-derivatives of tan -1 2 x, tan -1 2 y, tan -1 a; 2 , tan -1 y z . 

Ex. 2. Find the ^-derivatives of tan -1 4 £, tan -1 £*, tan -1 3 x 2 . 

Ex. 3. Show that the angles made with the x-axis by the tangents to 
the curve y = tan -1 x are 0°, 45°, and the angles between 0° and 45°. 

Ex. 4. Show how to determine the abscissas of the points of y = tan -1 x, 

the tangents at which cross the x-axis at an angle of 30°. 

2 x x 
Ex. 5. Find the ^-derivatives of the following : tan -1 , tan -1 , 

1 _ X 2' i + X 2' 



tan-* x , tan-i Vl + x2 - 1 , tan-i J-^. tan -i 3 « 2 x - x* 

y/\Z^& x >a + x a(a 2 -3x 2 ) 



74 DIFFERENTIAL CALCULUS. [Ch. IV. 

Ex. 6. (a) Show that D tan-i Jl^^t = 1 . (&) Show, by differenti- 
ation, that Z> ( tan -1 x + tan -1 - ) is independent of x. 
Ex. 7. Find anti-differentials of dx ' "''' 



1 + x 2 ' 1 + x 4 ' 1 + x 8 

52. Differentiation of cot -1 u. On proceeding in a manner simi- 
lar to that in Art. 51, it will be found that 

J- (co t-X„ ) = __l_||. 

If> = *, jLccot-i^-^. 

Ex. 1. Show, by means of the curve y = cot -1 x, that the derivative of 
cot -1 x is always negative. 

Ex. 2. Find the x-derivative of cot -1 - -f- log A/ . 

x ° *x + a 

53. Differentiation of sec 1 */. 

Put y = sec -1 w. 

Then sec y = u. 

On differentiation, sec y tan y -=- = — • 

QOT Or 3/ 

m dy _ 1 efat_ 1 cft^ 

' do; — sec y tan ?/ ota — sec y Vsec 2 ?/ — 1 ^' 

«.& A (sec -i^) = J=f*- (1) 

d& wv M 2-l da? y 

U u = x, then 4- (sec-i a?) = / • (2) 

Ex. 1. Explain the ambiguity of the result (2). Show that, when x is 
positive, the positive value of the radical is taken with the least positive 
value of sec -1 x. 

Ex. 2. Find the x-derivatives of sec -1 x 2 , sec -1 — , sec- 1 



x 2 - 1 

r 1 

Ex. S. Show by differentiation that tan -1 ~ 

independent of x. 



vT^x 2 Vl - x 2 



52-56.] DIFFERENTIATION OF FUNCTIONS. 75 

54. Differentiation of cosec -1 u. On proceeding in a manner 
similar to that in Art. 53, it will be found that 

4- (csc-i II) = ±= f?. (1) 

doc uVut-ldx 

If u = x, 4~ (csc-i 05) = (2) 

Ex. 1. Explain the ambiguity in sign in (2) by means of the graph of 
esc -1 u. Show that, when x is positive, the negative value of the radical is 
taken with the least positive value of esc -1 u. 

55. Differentiation of vers -1 u. 

Put y = vers -1 u. 

Then vers y = u. 

On differentiation, sin y — = 

dx dx 

dy _ 1 du _ 1 du 

dx sin y dx ^/± _ cos 2 y dx 

1 du 



Vl — (1 — vers y) 



2 dx 



i.e. ^- (vers-i u) = 1 f*. (1) 

doc V2u-u* d ™ 

If u = x, 4~ (vers-i oc) = 1 . (2) 

doc V2 03 - 058 

2 a** 2 

Ex. 1. Find the ^-derivative of vers -1 

1+x 2 

56. Differentiation of implicit functions : two variables. « 

N.B. Examples of the differentiation of implicit functions have been 
given in Exs. 13, 14, Art. 37. A preliminary study of these examples will 
help to make this article clear. 

Let y be an implicit function of x, the function y and the 
variable x being connected by a relation 

f(x,y) = e. (1) 



76 DIFFERENTIAL CALCULUS. [Ch. IV. 

If, as sometimes happens, it is impossible or inconvenient to 

express y as an explicit function of x, the derivative -^ may be 
obtained in the following way : 

On taking the cc-derivative of each member of (1), there is 
obtained a result of the form 

P+Q% = 0. (2) 

Fromtfcis | = -|- (3) 

Since the ^derivative of f(x, y) is P+ Q-, the differential of 

f(x, y) is (Art. 27) Pdx + Q^dx, i.e. (Art. 27) Pdx + Qdy. ■ 

ax 

dy 
Ex. 1. Find —-, when xy = c. 

Differentiation of the members of this equation gives y + x-y- = ; whence 

dv v du 

-— = — -. The x-derivative of xy is y + x^f; accordingly, the differential 

of xy is xdy + 2/d£. [Compare result (7), Art. 32.] 

Ex. 2. Write the differentials of the first members of the equations in 
Exs. 13, 14, Art. 37. 

Ex. 3. Find -— in each of the following cases : (i) x* + y* = a? ; 

(ii) x* + yi _ a s . (iii) ^_ 4. |_ = i ; (iv) (cos x)y - (sin y) x = 0. 

Ex. 4. Write the differentials of the first members of the equations in 
Ex.3. 

Note 1. It should be observed, as illustrated in Equation (2) and the 
above examples, that when the differential of f(x, y) is written Pdx + Qdy, 
P is the same expression as is obtained by differentiating /(cc, y) with respect 
to x, and at the same time regarding y as constant or letting y remain 
constant, and Q is the same expression as is obtained by differentiating 
f(x, y) with respect to y, and at the same time regarding x as constant or 
letting x remain constant. Here P is called the partial x-derivative of f(x, y), 
and Q is called the partial y-derivative of f(x, y). These partial derivatives 

are denoted by the symbols \ and ^ ' y) respectively. With this 

notation, result (3) may be written 

<fr_ & or s* n ' y) (4 \ 

*>- 3f(.x,yY 8 . w 

dv Tr n ' y) 



56.] DIFFERENTIATION OF FUNCTIONS. 77 

Ex. 5. In the exercises above, test the first statement made in this note. 

Note 2. Partial derivatives and the differentiation of implicit functions 
are discussed further in Chapter VIII. 

EXAMPLES. 

N.B. It is not advisable for the beginner to work the larger part of 
Exs. 1-8 before proceeding to the next chapter. Many of the differentiations 
required in these examples are far more difficult than those that are commonly- 
met in pure and applied mathematics ; but the exercise in working a fair 
proportion of them will develop a skill and confidence that will be a great 
aid in future work. 

Differentiate the functions in Exs. 1-4, 6, 7, with respect to x. 

1. (i) (2x-l)(3x + 4)(x 2 +ll); (ii) (« + £)(& + *); 
(iii) (a + a0»(6 + a;)»; (iv) ^4rfi (▼) - — - ; (vi) 



(x + &)» (1 + x)« Va 2 - x* 



(vii ) X ; (Viii) ^5+JL ; (ix) Vl+X 2 +Vl-^ . 

vl + x 2 Va + vx Vl + x 2 - Vl - x 2 

( X ) ( x Y. ( X i) x (a 2 + x 2 ) Va 2 - x 2 . 

\1 + Vl-x 2 / 



2. The logarithms of : (i) 7 x 4 + 3 x 2 - 17 x + 2 ; (ii) a T 2 ~ ^ 2 ; 

* a 2 + x 2 

(iii) 1 ±=; (iv) 'l±^inx. f ^ J vT+^+s 

a - Va 2 - x 2 >l-smx \ Vl + x 2 - x 

3. (i) sin 4 x 5 ; (ii) cos 2 7 x ; (iii)" sec 2 3 x ; (iv) tan (8 x + 5) ; 
(v) x m logx ; (vi) sini>x? ; (vii) sin nx • sin n x ; (viii) sin (sin x) ; 
(ix) sin (log nx); (x) log (sin nx). 



4. (i) log te=± - 1 ten-ia? ; (ii) log Jtans-1 
_25+l 2 w S \tanx + l 

(iii) log -J/i±*- 1 tan-ia. 
'1 — x 2 



5. Showthat D j x Va " + x2 + -log (x + Va 2 + x 2 ) } = Va 2 + a 2 . 

6. (i) tan^e 2 ; (ii) sin -1 (cos x); (iii) sin(cos- 1 x); 

(iv) tan-i (n tan x) ; (v) s i D -i 6 + «coss ; ^ e ax sin m rx; 

a + b cos x 

(vii) tan a x ; (viii) e x -v/^-^- 



78 DIFFERENTIAL CALCULUS. [Ch. IV. 

7. (i) (^p (ii) |e?; (iii) •; (iv) e* x ; (v) aj(^j (vi) (af)-. 

8. Find -^ under each of the following conditions : 

(i) ax 2 + 2 hxy + by 2 + 2 gx + 2fy + c = ; (ii) (x 2 + ?/ 2 ) 2 - a 2 (x 2 - y 2 ) =0 -; 
(iii) x 2 y* + sin y = ; (iv) sin (xy) = mx; (v) sin x sin y + sin xcosy~y; 
(vi) e» — e x + xy = ; (vii) X? = y x ; (viii) ye n » = ax m . 

9. Find -=£ in terms of x. when x = e » . 

dx 

10. Differentiate as follows : (i) 3 y 2 — 7 y + 11 with respect to, 3 y ; 
(ii) 4 £ 2 - 11 1 + 1 with respect to t + 2 ; (iii) a; with respect to sin a; ; 
(iv) sin z with respect to cos z ; (v) x with respect to Vl — x 2 . 

11. (i) Given y = '3n 2 -7u + 2 and w = 2x 3 + 3x + 2, find -^ ; (ii) given 

y = e s + s 2 and s = tan t, find -^ ; (iii) given v = V¥gs, s = £ gt 2 , find ~ 

du 
in two ways ; (iv) u = tan-^x*/), ?/ = e*, find — • 

12. Compute the angle at which the following curves intersect, and sketch 
the curves : (i) x 2 — y 2 = 9 and xy = 4 ; (ii) x 2 -\- y 2 = 25 and 4 y 2 = 9 a; ; 
(iii) y 2 = 8 (x + 2) and y 2 + 4 (x - 1) = ; (iv) y = 3 x 2 - 1 and y = 2 x 2 
+ 3 ; (v) x 2 + y 2 = 9 and (x - 4) 2 + y 2 - 2 y = 15. 

13. A point P is moving with uniform speed along a circle of radius a 
and centre ; AB is any diameter, and Q is the foot of the perpendicular 
from P on AB. Show that the speed of Q is variable, that at A and B it is 
zero, and at it is equal to the speed of P. (The motion of Q is called 
simple harmonic motion.') 

Suggestion : Denote angle AOP by 0, and OQ by x. Then x = a cos 6 ; 

hence ^ = -asin0^.1 
<fc <Z* J 

14. Suppose, in Ex. 13, the radius is 18 inches, and P is making 4 revolu- 
tions per second : what is the speed of Q when A OP is 15°, 30°, 45°, 60°, 
75°, 90°, 120°, 150°, respectively ? 



CHAPTER V. 

SOME GEOMETRICAL AND PHYSICAL APPLICATIONS. 
GEOMETRIC DERIVATIVES AND DIFFERENTIALS. 

57. The variation of functions, the sketching of graphs, and the 
determination of maxima and minima, which are discussed in Chapter 
VII., can be studied before entering upon this chapter. For some 
reasons it may be preferable to do this. 

58. This chapter gives some practical applications of the 
preceding principles of the calculus. The applications in Arts. 
59-62 are already familiar or obvious. The study of the geometric 
derivatives and differentials in Art. 67 is not of immediate im- 
portance, but will be found of more interest and value when 
Chapters XX. 5 XXV., are taken up. A glance over this article, 
however, will serve to make clearer and stronger the notions of 
a derivative and a differential. 

59. Slope of a curve at any point : rectangular coordinates. By 

the slope of a line (rectangular coordinates being used) is meant 
the tangent of the angle at which the line crosses the ie-axis. 
This angle is measured ' counter-clockwise - from the a>axis to the 
line, as explained in trigonometry. 

It has been shown in Art. 24 that at any point (x, y) on the curve 

y=f<&» (1) 

or 4>(a?,2/)=0. (2) 

The slope of the tangent is 

%' (*) 

The slope of the tangent drawn at a point on a curve is commonly 
called the slope of the curve at that point. 

79 



80 



DIFFERENTIAL CALCUL US. 



[Ch. V. 



TJie slope of the tangent (or the slope of the curve) at a particular 
point (x x , 2/j) is the number obtained by substituting (x^ y^ in the 
expression derived for (3) from (1) or (2). This slope is denoted 



by 



elasi 



(4) 




When the slope (4) is positive, the tangent crosses the ic-axis at 
an acute angle ; 

When the slope is negative, the tangent crosses the a?-axis at an 
obtuse angle ; 

When the slope is zero, the tangent is parallel to the a>axis ; 

When the slope is infinitely great, the tangent is perpendicular 
to the #-axis. These facts are illustrated in Fig. 12, in which 

the slope is positive at ^and P, 
negative at L and B, 
zero at M and Q, 
infinitely great at Fand S. 

Note. Symbol (4) does not mean ' the derivative of y± with respect to 
sci,' which is a meaningless phrase, since x\ and y\ are constants. 



EXAMPLES. 

1. Find the slope of the parabola 

4y=x* (1) 

at the points (xi, y{), (2, 1), (—3, f) ; and find the angles at which the tan- 
gents at the last two points cross the x-axis. 
(The student is supposed to draw the figure. ) 



60.] ANGLES AT WHICH TWO CURVES INTERSECT. 81 

dv x 
From (1), on differentiation, -^ = -. (2) 

This is a general expression, giving the slope of the curve at any point. 

From (2), on substitution, the slope at (#i, y{) (viz., ~ i ) =-^- 

\ dx\ j 2 

From (2), on substitution, the slope at (2, 1) = § = 1 ; 

accordingly, the tangent drawn at (2, 1) crosses the x-axis at the angle 45°. 

From (2), on substitution, the slope at (— 8, f) = -^— = — 1.5 ; 

accordingly, the tangent drawn at ( — 3, f ) crosses the a>axis at the angle 
123° 41.4'. 

2. Find the general expression giving the slope at any point on each of 
the curves in Art. 4, Ex. 3. 

3. Eeview the following examples : Ex. in Art. 24 ; Ex. 14 (6) in Art. 37 ; 
Exs. 5-8 in Art. 42 ; Exs. 3-6 in Art. 43 ; Ex. 2 in Art. 44 ; Ex. 3 in Art. 49, 
Exs. 3, 4, in Art. 51. 

4. Plot the following curves ; find the slope of each of them at the points 
described, and find the angle at which each of the tangents drawn to the 
curves at these points crosses the cc-axis : (i) the parabola y 2 = 8 a;, where 
x = 2, and where x = 8 ; (it) the parabola x 2 = 8 y, where x = 8 ; (iii) the 
circle x 2 + y 2 = 13 at (2, 3) ; (iv) the circle x 2 + y 2 = 18 at (3, 3) ; (v) the 
curve 3 y 2 = x s at (3, 3) ; (yi) the curve 3 y 2 =(x+ l) 3 at (2, 3) ; (yii) the hy- 
perbola x 2 — y 2 = 20 at (6, 4) ; (viii) the hyperbola xy = 24 at (6, 4) . 

60. Angles at which two curves intersect. By the angle (or 
angles) at which two curves intersect is meant the angle (or angles) 
formed by the tangents drawn to each of them at their point (or points) 
of intersection. 

By the angles of intersection of a straight line and curve is 
meant the angles between the line and the tangents drawn to the 
curve at the points of intersection. 

The method of finding the angles of intersection of two curves, as illus- 
trated in the following examples, may be outlined thus : 

1. Find the points of intersection of the carves ; 

2. Find the slope of each curve at each of these points ; 

thence can be obtained the angles at which the tangents drawn at these 
points cross the x-axis. 

3. From either the slopes or the angles just described, find the angle 
between the tangents at each point of intersection. 



82 



DIFFERENTIAL CALCULUS. 



[Ch. V. 




Fig. 13. 



EXAMPLES. 

1 . Find the angles at which the circle 
x 2 + y 2 = 72 and the parabola y 2 = 6 x 
intersect. These curves and the tan- 
gents concerned are shown in Fig. 13. 

On solving the equations of the 
curves simultaneously, the points of 
intersection are found : viz. , 

P(6, 6) andP(6-6). 



The method of last article applied to each curve at P brings out the 
following results : 

Slope of PT X (i.e. tan X7\P) = \ ; whence XT X P = 26° 33.9'. 

Slope of PT 2 (i.e. tan XT 2 P)=- 1 ; whence XT 2 P = 135°. 

.-. T X PT 2 = XT 2 P - XT X P = 135° - 26° 33.9' = 108° 26.1', 



and thus, 



7VPP = 71 33.9' 



In a similar manner the angle of intersection at B will be found to have 
the same value, as is also apparent from the symmetry of the figure. 

The angle of intersection may also be found directly from the slopes of 
PTi and PT 2 , for 

tan XT 2 P - tan XT X P 



tan T X PT 2 = tan (XT 2 P - XT ± P) = 

= - 1 -* =-8. 

H(-lxl) 



1 + tan XT 2 P • tan XT X P 



.'. T 1 PT 2 = 108° 26.1'. 



x + 6 inter- 



2. At what angles does the line 
sect the parabola 2 y = x 2 ? 

The line, parabola, and tangents concerned are 
shown in Fig. 14. On solving the equations of 
the line and the parabola simultaneously, it is _£ 
found that 




at P, x=- 2.6056; at Q, x 
dy 



4.6056. 



Fig. 14. 



From 2 y = x 2 , it follows that ^ = x ; this is the slope of the parabola 

dx 

WPQ at any point (x, y). 

.-. slope of PTi =- 2.6056 ; whence X7\P = 110° 59.8'; 
slope of QT 2 = 4.6056 ; whence XT 2 Q = 77° 45'. 



61.] EQUATION OF THE TANGENT. 83 

Now, slope of SV = 1 ; whence XSV = 45°. 

.-. 8PT X = XZ\P- XSV = 65° 59.8'; 
SQT 2 = XT 2 Q - XSV = 32° 45'. 
3. Keview Exs. 23, 24, Art. 37, and Ex. 12, Art. 56. 

61. Equations of the tangent and the normal drawn at a point on 
a curve. 

In Fig. 15, Art. 62, P is the point (x 1} y-^ on the curve y =f(x) ; 

PT is the tangent which touches the curve at P\ 

PN, drawn at right angles to PT, is the normal to the curve at P. 

The slope of the tangent PT = ^l [Art. 59 (4)]. (1) 

ClXi 

It is shown in analytic geometry that if the slope of a line is 

m, the slope of a line perpendicular to it is Accordingly, 

m 

the slope of the normal PN=-^> (2) 

dy 1 

It is shown in analytic geometry that the equation of a line 

which passes through a point (x 1} y^) and has a slope m is 

y — y 1 = m(x — x l ). 

Accordingly, since PT passes through P(x 1} y^ and has the slope 

(i), 

the equation of the tangent at (x v y^), is y—y 1 = — -i (x — Xj). (3) 

CtX-t 

Since PA 7 " passes through P(^, y^) and has the slope (2), 

the equation of the normal at (x v y x ) i^y—y 1 = — — — v - (x — x t ) (4) 

uy^ 

EXAMPLES. 

1. Write the equations of the tangents and normals to the circle and 
parabola at P(6, 6) in Fig. 13. 

At P, (see Ex. 1, Art. 60), slope of PPi = \. 

.-. equation of tangent P7\ of the parabola is y — 6 = \(x — 6) ; 
and the equation of the normal to the parabola at P is y — 6 = — 2(x — 6). 

These equations reduce to 2 y — x = 6, 
and ?/ + 2 x = 18, respectively. 

2. Find the equations of the tangents and normals drawn to the circle and 
parabola at B in Fig. 13. 



84 



DIFFERENTIAL CALCULUS. 



[Ch. V. 



3. Write the equations of the tangents to the parabola at P and Q in 
Fig. 14 ; also the equations of the normals at these points. 

Find the lengths of 02\ and OT 2 . 

4. Write the equations of the tangents and normals for each of the curves 
and points appearing in Ex. 4, Art. 59. 

62. Lengths of tangent, subtangent, normal, and subnormal, for 
any point on a curve : rectangular coordinates. Let P be a point 
(x lf ft) on the curve y=f(x) [or, <f>(x, y) = 0]. 

At P let the tangent PT be drawn ; likewise the normal PN 
and the ordinate PM. The length of the line PT, namely, that 
part of the tangent which is intercepted between P and the cc-axis, 

is here termed the length of the tan- 
gent. The projection of TP on the 
a>axis, namely TM, is called the 
subtangent. The length of the line 
PJV, the part of the normal which 
is intercepted between P and the 
aj-axis, is termed the length of the 
normal. The projection of PN on 
the #-axis, namely MN, is called 
the subnormal. 

Note 1. The subtangent is measured from the intersection of the tangent 
with the x-axis to the foot of the ordinate ; the subnormal is measured from 
the foot of the ordinate to the intersection of the normal with the se-axis. 

A subtangent extending to the right from T is positive, and one extending 
to the left from T is negative ; a subnormal extending to the right from M is 
positive, and one extending to the left from M is negative. 

Let angle XTP be denoted by a; then tana = -^« In the 

ax± 

triangle TPM: MP = y 1 \ TM = ft cot a = yj^ *; TP=y 1 csca 

dy l 

=2 "V 1+ (SJ i ( or ' ^-vs^Tra'-*Vi+(D} In 

the triangle PMN: angle MPN= a ; MN= y 1 tan MPN= ft^i; 




PJST=y 1 sec MPN 



= Vi\] 



1 + 



ytV 1 + /r * Y 






or, PJST=^MP 2 + MN 2 



dx 1 



62.] LENGTHS OF TANGENT, ETC. 85 

It being understood that y and -^ denote the ordinate and the 

dx 

slope of the tangent at any point on the curve, these results may 

be written : 

subtangent = y-^; 
dy 

subnormal = y—&; 
doc 



length of tangent = y^jl + i^)'. 



\dyl 



length of normal = y^l + (^V 



Note 2. It is better for the student not to use these results as formulas, 
but to obtain the lengths of these lines in any case directly from a figure. 

EXAMPLES. 

N.B. Sketch all the curves and draw all the lines involved in the follow- 
ing examples. 

1. In each of the following curves write the equations of the tangent and 
the normal, and find the lengths of the subnormal, subtangent, tangent, and 
normal, at any point (sti, y±), or at the point more particularly described : 
(1) Circle x 2 + y 2 = 25 where x= — 3; (2) parabola y 2 = Sx at x = 2; 
(3) ellipse b 2 x 2 + a 2 y 2 = a 2 b 2 ; (4) sinusoid y = sin x ; (5) exponential curve 
y = e x . 

2. Where is the curve y(x — 2) (x — 3) = x — 7 parallel to the x-axis ? 

3. What must a 2 be in order that the curves 16 x 2 + 25 y 2 = 400 and 
49 x 2 + a 2 y 2 = 441 intersect at right angles ? 

X 

4. In the exponential curve y = be a show that the subtangent is constant 

and that the subnormal is ^- • 
a 

5. In the semi-cubical parabola Sy 2 =(x + l) 3 show that the subnormal 
varies as the square of the subtangent. 

6. In the hypocycloid of four cusps, x* 4- y* = as : (1) Write the equa- 
tion of the tangent at (xi, y{) ; (2) show that the part of the tangent inter- 
cepted between the axes is of constant length a ; (3) show that the length 
of the perpendicular from the origin on the tangent at (x, y) is Vaxy ; (4) if 
p, p\ be the lengths of the perpendiculars from the origin to the tangent and 
normal at any point on the curve, 4p 2 4- pi 2 = a 2 . 



86 DIFFERENTIAL CALCULUS. [Ch. V. 

7. In the parabola x? + y? = a^, write the equation of the tangent at 
any point (xi, ?/i), and show that the sum of the intercepts made on the axes 
by this tangent is constant. Show that this curve touches the axes at (a, 0) 
and (0, a). 

8. In the cycloid x = a(0 — sin 0), y = a(l — cos 0) : (1) Calculate the 
lengths of the subnormal, subtangent, normal, and tangent at any point 
(£> V) ; ( 2 ) show that the tangent at any point crosses the y-axis at the angle 

a 

-; (3) show that the part of the tangent intercepted between the axes is 

6 f) 

a0cosec — 2 a sec-. [See Art. 35.] 

9. In the hyperbola xy = c 2 : (1) Show that for any point (x, y) on 

the curve the subnormal is — 2L and the subtangent is — x ; (2) find the 

c 2 
x- and ^-intercepts of the tangent at any point (xi, yi), and thence deduce a 
method of drawing the tangent and normal to the curve at any point on it. 
Show that the product of these intercepts is 4 c 2 . 

10. In the semi-cubical parabola ay 2 = x 3 , show that the length of the 
subtangent for any point (x, y) is § x ; thence deduce a way of drawing the 
tangent and the normal to the curve at any point on it. 

Q 

11. Show that the parabola x 2 = 4 y intersects the witch y = 

at an angle tan" 1 3 ; i.e. 71° 33' 54". x 2 + 4 

12. Find at what angles the parabola y 2 = 2 ax cuts the folium of Descartes 
x z + y s = 3 axy. 

13. In the curve x m y n = a m+n show : (1) That the subtangent for any 
point varies as the abscissa of the point ; (2) that the portion of the tangent 
intercepted between the axes is divided at its point of contact into segments 
which are to each other in the constant ratio m : n ; (3) thence, deduce a 
method of drawing the tangent and the normal at any point on the curve. 
(The curves x m y n = a m+n , obtained by giving various values to m and w, are 
called adiabatic curves. Instances of these curves are given in Exs. 9, 10, 
and in the parabolas in Exs. 11, 12.) 

14. Show that all the curves obtained by giving different values to n in 
2, touch one another at the point (a, 6). Draw the curves in 



(fN!)' 



which (a, b) is (4, 7), n = 1, n = 2. 

15. Show that the tangents at the points where the parabola ay = x 2 
meets the folium of Descartes x 3 + y 3 = 3 axy are parallel to the x-axis, and 
that the tangents at the points where the parabola y 2 = ax meets the folium 
are parallel to the y-axis. Make figures for the curves in which a = 1 and 
a = 4. 




63.] SLOPE OF A CURVE AT ANY POINT. 87 

63. Slope of a curve at any point : polar coordinates. Let CM 

be a curve whose equation is 
r=f(0), [or <j>(r, 0)=O], and 
P be any point on it having 
coordinates r 15 1; with reference 
to the pole and the initial 
line OL. Draw OP; then 
OP=r 1 , and angle LOP=0 V 
Through P and Q (a neigh- 
bouring point on the curve), 

draw the chord TPQ, and draw OQ. From P draw PR at right 
angles to OQ. 

Let angle POQ = A0 l5 and OQ = r x + Ar x ; 

then Pi? = 7-j sin A0 l5 and PQ = i\ + A^ — r 2 cos A0j. 

The angle between the radius vector drawn to any point P and 
the tangent at P is usually denoted by if/. Since 

if/ = lini A .^o angle RQP, 

then, using the general coordinates r, 0, instead of r 1} h 

EP 



tan i/r = lim Afl:y) . 



= lim 



r sin A0 



±e±o 



Ar — r cos A0 



On replacing cos A0 by its equal, 1 — 2 sin 2 -§■ A0, and dividing 
numerator and denominator by A0, this becomes 



tan if/ = lim A0=y) 



sin A 6 

r ;r~ 

A0 


> 


r 


At* 

— +rsin-i-A0. 

tanx|/ = ^. 
dr 


sin £ A0 
iA0 


dr 
d$ 



That is, tanx|/ = ^^. (1) 

dr 

The angle between the initial line and the tangent at P is 
usually denoted by </>. 



88 DIFFERENTIAL CALCULUS. 

It is apparent from Fig. 17 that 

<|> = t|/ + e. 



[Ch. v. 



(2) 




Note. Results (1) and (2) are true for all polar curves, whatever the 
figure may be. The student is advised to draw various figures. 

64. Lengths of the tangent, normal, subtangent, and subnormal, for 
any point on a curve : polar coordinates. 

In Fig. 18 is the pole and OL is the initial line. At P any 
point (?*!, 0^, on the curve CR, whose 
equation is r=f($), [or <f>(r, 0) = O], 
let the tangent PT and the normal 
PN be drawn. Produce them to 
intersect NT, which is drawn through 
at right angles to the radius vector 
OP. 

The length of the line PT is termed 
the length of the tangent at P; the 
projection of PT on NT, namely OT, 
is called the polar subtangent for P; 
the length of PN is termed the 
length of the normal at P; the projec- 
tion of PN on NT, namely ON, is called the polar subnormal for P. 

Note. In Art. 59 the line used with the tangent and the normal is the 
cc-axis. Here the line so used is not the initial line, but the line drawn 
through the pole at right angles to the radius vector of the point. 




Fig. 18. 



In the triangle OPT : 
OT 



OP tan OPT 



64.] LENGTHS OF TANGENT, ETC. 89 

i.e. (on removing the subscripts from the letters) 
polar subtangent = r tan if/ = r% — ; 
also, TP= OP sec OPT; 

i.e. polar tangent length = r sec \f/ = ryl + r^( ^- J • 

[ft-: rp= V5p« + oF =^ + r*(f J= r> /l + HgJ.] 

In the triangle OP^T : 

angle NPO = 90 -^; 

0>= OP tan JVPO; 
i.e. polar subnormal = r cot ^ = ~ ; 

also, JVP = OP sec JVPO; 

l".e. polar normal length = r cosec ^ =yr® + f^\ . 



Or : NP = Vop 2 + Oi^ 2 = yf* + 7— 

df) 

Note. In Fig. 18 r increases as increases ; accordingly — is positive, 

dd dr 

and hence the subtangent is positive. Thus when — is positive, the sub- 

dr 

tangent is measured to the right from an observer at looking toward P. 

df) 

When r decreases as 6 increases, and thus — is negative, the subtangent is 

dr 

measured to the left of the observer looking toward P from 0. The student 

is advised to construct figures for the various cases. 



EXAMPLES. 

tf.B. In the following examples make figures, putting a = 4, say. Apply 
the general results found in these examples to particular concrete cases, e.g. 

a = 6 and 6 — ^, a = 2 and 6 = — , etc. The angle 0, as used in the equa- 
tions of the curves, is expressed in radians. 



90 DIFFERENTIAL CALCULUS. [Ch. V. 

1. In the following curves calculate the lengths of the subnormal, sub- 
tangent, normal, and tangent, at any point (r, 0) : (1) The spiral of 
Archimedes r = a6 ; (2) the parabolic spiral or lituus r 2 = a 2 8 (i.e. 
r = ad?) ; (3) the hyperbolic spiral (or the reciprocal spiral) rd = a; 
(4) the general spiral r = ad n . (The preceding spirals are special cases 
of this spiral.) 

2. From the results in Ex. 1 deduce simple geometrical methods of 
drawing tangents and normals to the spirals in (1), (2), (3). 

3. Do as in Exs. 1, 2, for the logarithmic spiral r = e ae . In this 
curve each of the lengths specified varies as the radius vector. 

4. (a) In the spiral of Archimedes r = ad, show that tan \p = 6. Find 
^ and 4> in degrees when angle TOP (Fig. 17) = 40°, and when TOP = 70°. 
(&) In the curve r = 4 0, find \p and when r = 2. 

5. (a) In the logarithmic spiral r = ce ae , show that \p is constant. 
This spiral accordingly crosses the radii vectores at a constant angle, and 
hence is also called the equiangular spiral, (b) Show that the circle is a 
special case of the logarithmic spiral, and give the values of ^ and a for 
this case. 

Q 

6. In the parabola r = asec 2 -, show that + \p = ir. Make a prac- 

u 

tical application of this fact to drawing tangents and normals of this curve. 

On a 

7. In the cardioid r = a (1 — cos 0), show that = — , ^ =-, sub- 

6 8 

tangent = 2 a tan - sin 2 — Apply one of these facts to drawing the tangent 

and normal at ?■ point on the curve. 

65. Applications involving rates. Applications of this kind 
have already been made in Arts. 26, 37. Rates and differentials 
have been discussed in Arts. 25-27. It has been seen, Art. 26, 
Eq. (1), that if y =f(x), then 

dt J V * } dt dx dt 

In words, the rate of change of a function of a variable is equal 
to the product of the derivative of the function with respect to 
the variable and the rate of change of the variable. The following 
principles, which are proved in mechanics, will be useful in some 
of the examples : (a) If a point is moving at a particular moment 

in such a way that its abscissa x is changing at the rate — , and 



65. j APPLICATIONS INVOLVING BATES. 91 

its ordinate y is changing at the rate -^, and if — denote its rate 

a at at 

of motion along its path at that moment, then 

\dtj \dtj [dtj 

(6) If a point is moving in a certain direction with a velocity 
v, the component of this velocity in a direction inclined at an 
angle a to the first direction, is v cos a. 

For instance, if a point is moving so that its abscissa is increasing at the 
rate 2 feet per second and its ordinate is decreasing at the rate 3 feet per 
second, it is moving at the rate V2 2 + 3 2 , i.e. V13 feet per second. Again, 
if a point is moving at the rate of 6 feet per second in a direction inclined 
60° to the x-axis, the component of its speed in a direction parallel to the 
x-axis is 6 cos 60°, i. e. 3 feet per second, and the component parallel to the 
y-axis is 6 cos 30°, i.e. 5.196 feet per second. 



EXAMPLES. 
N.B. Make figures. 

1. If a particle is moving along a parabola y 2 = 8 x at a uniform speed of 
4 feet per second, at what rates are its abscissa and its ordinate respectively- 
increasing as it is passing through the point (x, y) and x has successively the 
values 0, 2, 8, 16 ? 

2. A particle is moving along a parabola y 2 = 4 x, and, when x = 4, its 
ordinate is increasing at the rate of 10 feet per second : find at what rate its 
abscissa is then changing, and calculate the speed along the curve at that 
time. 

3. A particle is moving along the hyperbola xy = 25 with a uniform speed 
10 feet per second : calculate the rates at which its distances from the axes 
are changing when it is distant 1 unit and 10 units respectively from the 
y-axis. 

4. A vertical wheel of radius 3 feet is making 25 revolutions per second 
about an axis through its centre : calculate the vertical and the horizontal 
components of the velocity, (1) of a point 20° above the level of the axis; 
(2) of a point 65° above the level of the axis. 

5. A point is moving along a cubical parabola y = x 3 : find (1) at what 
points the ordinate is increasing 12 times as fast as the abscissa ; (2) at what 
points the abscissa is increasing 12 times as fast as the ordinate ; (3) how 
many times as fast as the abscissa is the ordinate growing when x = 10 ? 



92 DIFFERENTIAL CALCULUS. [Ch. V. 

66. Small errors and corrections : relative error. 

If 2/ =/(*), (1) 

then by Art. 21, dy =/'(») • dx, (2) 

in which dx is an assigned change in x. It has been seen (Note 
3, Art. 27) that dy is approximately the change in y due to dx. 
An important practical application may be made of this principle. 
For it follows that if dx be regarded as a small error in the 
assigned or measured value of x, then dy is an approximate value 
of the consequent error in y. 

The ratio ^ or £M . dx (3) 

is, approximately, the relative error or the proportional error, i.e. 
the ratio of the error in the value to the value itself. 

The approximate values of the correction and relative error may also be 
deduced from the theorem of mean value. For, if y = f(x), and Ax be an 
error in x, then f(x -f Ax) — /(x) is the error in y, i.e. the correction that 
must be applied to y. Now by (3) Art. 108, on putting a = x and h = Ax, 

f(x + Ax) - f(x) =f'(x + 0- Ax) ■ Ax. 

Hence, on denoting the error in y by Ay, 

Ay =f'(x) • Ax approximately. 

Aw f r (x) 
From this the relative error is, approximately, — = v y • Ax. (4) 

EXAMPLES. 

1. The side a of a square is measured, but there is a possible error 
Aa : find approximately the error in the calculated value of the area. Let 
A denote the area. Then A = a 2 ; whence A A = 2 a • Aa approximately. 

2. If the measured length of the side is 100 inches and this be correct 
to within a tenth of an inch, find an approximate value of the possible error 
in the computed area, and an approximate value of the relative error. 

In this case, approximately, Aa = 2 x 100 x . 1 = 20 square inches. The 

20 1 

relative error is, approximately, or — ; that is, 20 square inches in 

10,000 square inches, or 1 square inch in 500 square inches. 



66,66 c] APPLICATIONS TO ALGEBRA. 93 

3. A cylinder has a height h and a radius "r inches ; there is a possible 
error A?' inches in r : find by the calculus an approximate value of the possible 
error in the computed volume. If h = 10 inches and the radius is 8 ± .05 
inches, calculate approximately the possible error in the computed volume 
and the relative error made on taking r = 8 inches. 

4. Find approximately the error made in the volume of a sphere by 
making an error Ar in the radius p. The radius of a sphere is said to be 20 
inches : give approximate values of the errors made in the computed surface 
and volume, if there be an error of .1 inch in the length assigned to the radius. 
Also calculate the relative errors in the radius, the surface, and the volume, 
and compare these relative errors. 

5. Two sides of a triangle are 20 inches and 35 inches. Their included 
angle is measured and found to be -48° 30'. It is discovered later that there 
is an error of 20' in this measurement. Find, by the calculus, approximately 
the error in the computed value of the area of the triangle. Compare the 
relative errors in the angle and in the area. 

6. The exact values of the errors in the computed values in Exs. 1-4 
happen to be easily found. Calculate these exact values, and compare with 
the approximate values already obtained. 

7. (1) Two sides, a, &, of a triangle are measured, and also the included 
angle C: show that the approximate amount of the error in the computed 
length of the third side c due to a small error AC made in measuring O, is 

ab sin C ^~ 



Va 2 + b' 2 — 2 ab cos C 



(2) Calculate the approximate error in the computed value of the third 
side in Ex. 5. 

66 a. Applications to algebra. Solution of equations having 
multiple roots. 

The following properties are shown in algebra : 
(a) If a is a root of the equation f(x) = 0, 

then x — a is a factor of the expression f(x) ; 
and conversely, 

if x — a is a factor of the expression f(x), 
then a is a root of the equation f(x) = 0. 



94 DIFFERENTIAL CALCULUS. [Ch. V. 

(b) If a is an r-fold (or r- tuple) root of the equation f(x) = 0, 
then (x — a) r is a factor of the expression f(x) ; 
and conversely, 

if (x — a) r is a factor of the expression f(x), 

then a is an r-fold (or r-tuple) root of the equation f(x) = 0. 

E.g. the equation x 3 — 7x? + 16x — 12 = 
has roots 2, 2, 3. 

The equation may be written (x — 2) 2 {x — 3) = CL 

The roots of the equation x 3 — 7 x 2 + 16 a? — 12 = are 2, 2, 3 ; 
the factors of the expression x* — 7x 2 -\-16x — 12 are (x — 2) 2 , x — 3. 

Note. When a number is a root of an equation more than once (e.g. 
the number 2 in the equation above) , it is said to be a multiple root of the 
equation. If an equation has r roots equal to the same number, the number 
is said to be an r-fold or an r-tuple root of the equation. 

Theorem^.. If f(x) is a rational integral function of x, and 
(x — a) r is a factor of f(x), then (x — a) r_1 is a factor of f'(x). 

For, let f{x) = (x — a) r $(x). 

Then f(x) = r(x- a)*" *<£ (x) -j- (x - a) r <f>'(x) 

= (x — a)^ 1 [><£ (x) + (x — a)<f>'(x)']. 

Accordingly, (x — a) r ~ l is a part of the Highest Common Factor 
of /(a;) and /'(a). 

Also, if (x-ay- 1 is a part of the H.C.F. of f(x) and f(x), 
(x — af is a factor off(x). 

Prom Theorem A and property (6) there follows : 

Theorem B, If f(x) is a rational integral function ofx, and a is 
an r-tuple (or r-fold) root of the equation f(x) = 0, then a is an 
(r — l)-tuple root of the equation f '(x) = 0. 

It follows from Theorems A and B that if the equation f(x) = 



67.] 



G EOMETBIC DERIVA TIVES. 



95 



has multiple roots, they will be revealed on finding the H. C. F. 
of f(x) and /(a?). 

Ex. 1. Solve x 3 — 2 x 2 — 15 x + 36 = (a) by trying for equal roots. 
The derived equation is 3 x 2 — 4 x — 15 = 0. (b) 
The H. C. F. of the first members of these equations is x — 3. 
Accordingly (x — 3) 2 is a factor of the first member of (a). 
Hence, as found on division by (x — 3) 2 , (a) may be written 

(x - 3)2(b + 4) = ; 

and thus the roots of (a) are 3, 3, — 4. 

Ex. 2. Solve the following equations : 

(1) 3x 3 +4x 2 -x-2 = 

(2) 4x 3 + 16x 2 + 21^ + 9 = 

(3) a 4 - 11 £ 3 + 44z 2 - 76 £ + 48 = 

(4) 8z 4 + 4z 3 -62z 2 -61x- 15=0 

(5) z 5 + z 4 - 13z 3 - z 2 -f48x- 36=0. 

Ex. 3. Eind the condition that x n — px 2 + r = may have equal roots. 

N.B. It is better to postpone the reading of the larger part of Art. 67 
until the topics in it are required, or referred to, in the integral calculus. 



67. Geometric derivatives and differentials. 



(a) Derivative and differential of an 
area : rectangular coordinates. Let PQ 

be an arc of the curve y =f(x). Take 
any point on PQ, V(x, y) say, and take 
T(x + Ax, y + Ay). Construct the rec- 
tangles VX and TM as shown in Eig. 19. 
Draw the ordinate BP, and let the area of 
BPVM be denoted by A ; then the area 
of M VTN may be denoted by &A. 

Now, 




Fig. 19. 



rectangle VN < MVTN< rectangle MT 
i.e. y • Ax < AA < (y + Ay) Ax. 

AA 

Hence, on division by ax, y < — < y 4- Ay. 

Ax 



(1) 



96 DIFFERENTIAL CALCULUS. [Ch. V. 

On letting Ax approach zero, these quantities (Arts. 18, 22, 23) approach 

dA 

the values y, — , y, respectively. 
dx 

That is, the derivative of the area BPVM with respect to the abscissa 
x of V, is the measure of the ordinate of V. On denoting this measure by y, 
result (2) means (Art. 26) that the area BPVM is increasing y times as fast 
as the abscissa of V. From (2) it follows by Art. 27 that 

dA = y . doc. (3) 

That is, the differential of the area BPVM is the area of a rectangle 
whose height is the ordinate M V and whose base is dx, the differential of the 
abscissa of V. 

Ex. 1. Find the derivative of the area between the x-axis and the curve 
y = x 3 , with respect to the abscissa : (a) at the point whose abscissa is 2 ; 
(b) at the point whose abscissa is 4. 

(a) *A=% (where x = 2,) = 2 3 = 8. (4) 

dx 

(6) — = ?/, (where x = 4,) = 4 3 = 64. (5) 

These results mean that, if an ordinate, like VM in the figure, is moving 
to the right or left at a certain rate, the area of the figure bounded on one 
side by that ordinate is changing, in case (a) at 8 times that rate, and in 
case (Z>) at 64 times that rate. 

Ex. 2. Find the differentials in Ex. 1 (a) and (5), when dx = .1 inch. 
Show these differentials on a drawing. 

By (3), (4), and (5), in case (a), dA = .8 square inch; in case (6) 
dA = 6.4 square inches. 

Note. The area .8 square inch is nearly the actual increase in area 
between the curve and the aj-axis when the ordinate moves from x — 2 to 
£C = 2.1 ; and 6.4 square inches is nearly the increase in this area when the 
ordinate moves f rom ~x = 4 to' # = 4.1. These increases are calculated in 
Ex. 16, Art. 111. 

It is evident that the smaller dx is taken, the more nearly will the differen- 
tial of the area become equal to the actual increase of the area between the 
curve and the x-axis. 

Ex. 3. Show that the ^/-derivative of an area between the curve and the 
y-axis is x. Thence deduce that the ^-differential of this area is x dy, and make 
a figure showing this differential area. 



67.] 



GEOMETRIC DERIVATIVES. 



97 



Ex. 4. In the case of the cubical parabola y = x s find — and — ; then 

dx dy 

calculate the differential of the area between this curve and the z-axis at the 
point (2, 8) , taking dx = .2. Also calculate the differential of the area between 
this curve and the y-axis at the same point, taking dy = .2. Show these 
differentials in a figure. 



(6) Derivative and differential of an area : polar coordinates. Let 

PQ be an arc of the curve /(r, 8) = 0. On 
PQ take any point F(r, 0), and take the 
point TFO'+Ar, + A0). About describe 
a circular arc VN intersecting OW in JV, and 
describe a circular arc WM intersecting OV 
in M. Then XW=Ar, and VOW = Ad. 
Also (PI. Trig., p. 175), area sector VOX = 
i r 2 A0, and area sector MOW= \ (r+Ar) 2 A0. 

Draw OP. Let the area of POV be 
denoted by A ; then the area of VOW may 
be denoted by AA. 

Now, area VON < area VO W < area MO W ; 

i. e. I r 2 A8 <AA<l(r + Ar) 2 A0. 

•'• ir2< if <Kr + Ar)2 - 

On letting A0 approach zero, these quantities (Arts. 18, 22, 23) approach 
the values . . ^j_ 




2 dd' 2 



respectively. 






(i) 



Eesult (1) means that, if the radius vector is revolving at a certain rate 
the area passed over by the radius vector, when its length is r, is increasing 
at a rate which is \ r 2 (i.e. the number) times the rate of revolution. 

It follows from (1) and Art. 27 that 

1 



dA 



r'ldQ. 



(2) 



Ex. 5. Show that in the case of the circle the differential of the area swept 
over by a revolving radius is the additional area passed over. 

Ex. 6. In the spiral of Archimedes r = 2 8 find the derivative of the area 
swept over by the radius vector, with respect to 8. Calculate the differential 
of this area when : (1) 8 = 30° and dd = 30' ; (2) r = 2 and dd = 1°. Make a 
figure showing these differentials. 

Ex. 7. In the cardioid r = 4(1 — cos 8) find the ^-derivative of the area. 
Calculate the differential of the area when : (1) = 60° and dd=:l ; (2) 8 = and 
dd = 2° ; (3) 8 = 330° and dd = 1°. Make a figure showing these differentials. 



98 



DIFFERENTIAL CALCULUS. 



[Ch. V. 



(c) Derivative and differential of the length of a curve : rectangular 
coordinates. Let PQ be an arc of the 

curve y=f(x). On PQ take any point 
M(x, y), and take the point N(x + Ax, 
y + Ay) ; and draw the chord MN. On 
denoting the length of the arc PM by s, 
the length of the arc MN may be denoted 
by As. 

When Ax approaches zero, chord MN 

~ and arc MN approach equality. It can be 

Fig. 21. shown rigorously (see Inf. Gal., p. 102) that 




limAx=o 



arc MN 
Ax 



limA*=i=o 



chord MN , 
Ax '' 



As .. V(Ax) 2 +(A?/) 2 .. 

hniAx=o -— = liniAx=o — - — — = hmAx=o 

Ax Ax 



M%i 



That is, 



ds 
doc 






From (2), (3), and Art. 27, 
ds 



<w 



doc; 



dy. 



(1) 



(2) 



(3) 



(4) 



(5) 



Ex. 8. Show that for a given dx and the actual derivative -=^ at M, the 

second member of (4) gives the length of the intercept of the tangent, 
namely, MT. Show that for a given dx, and using dy to denote the exact 
corresponding change in the ordinate, the second members in (4) and (5) 
give the length of the chord of the arc, namely, the line MN. 

Note. It is shown in Art. 137 how to find the length of the arc MN 
corresponding to an increment dx in x. The smaller dx is, the more nearly 
will MT, arc MN, and chord MN, become equal to one another. See Ex. 6, 
Art. 19. 

Ex. 9. (1) Calculate the x-derivative and the ^-derivative of the arc 
of the parabola y 2 = 4 ax. (2) Find the x-derivative of the hypocycloid 
at + y i - a f 

Ex. 10. In the cubical parabola y = x 3 calculate the differential of the arc 
at the point (2, 8) when : (1) dx = .2 ; (2) dy = .1. Show these differen- 
tials in a figure. (The actual increments of the arcs can be computed by 
Art. 209.) 



67.] 



GEOMETRIC DERIVATIVES. 



99 



(d) Derivative and differential of the ^ 
length of a curve : polar coordinates. 

Let PQ be an arc of the curve /(r, d) = 0. 
On PQ take any point F(r, 0), and take 
W(r + Ar, 6 + A0). Denote the length of PV 
by s ; then the length of VW may be denoted 
by As. Draw the chord VW. 
Now, as in (c) , 



lim 



A0=O 



ar c VW 

Ad 



(i.e. *) 



lim 



A0=O 



chord FTF 




A0 



(1) 



Fig. 22. 



About describe a circular arc VM intersecting OW in M, and draw VT 
at right angles to IF. Then angle VO W = Ad, and M W = Ar. 

.-. TIF = OTT— OT = r + Ar —V cos A0, and VT = r sin Ad. 



chord FTF = V ( VT) 2 + ( T>F) 2 = V(r sin A0)" 2 + [r(l - cos Ad) + Ar] 
chord VW 



Ad 



V( 



-^M; 



sin ^ M .siniA0 + ^ 
JA0 2 A0_ 



(2) 



™ A9i0 



cho rd FEF 
A0 



++(%)' 



since, if A0 = 0, ^^ = 1, sm ? A ^ = 1, and sin J Ad = 0. 

' A0 %Ad * 



Hence, by (1), 



cze 



V"+{S)' 



A0 



(3) 



On multiplying each member of (2) by — , and then letting Ad, and con- 

Ar 



sequently Ar, approach zero, it will be found that 



From (3), (4), and definition Art. 27, 



Wey 



and 



cfs 



=VW 






+ 1 • dr. 



(4) 

(5) 
(6) 



Ex. 11. Find the derivative of the arc of the spiral of Archimedes r— ad: 
(1) with respect to the angle ; (2) with respect to the radius vector. 

Ex. 12. Calculate the differential of the arc of the Archimedean spiral 
r = 2 d when d = 2 radians and dd — 1°. Make a figure. (The actual incre- 
ment of the arc can be computed by Art. 210.) 



100 



DIFFERENTIAL CALCULUS. 



[Ch. V. 



(e) Derivative and differential of the volume of a surface of revolu- 
tion. Let PQ be an arc of the curve y =f(x). On FQ take any point 

L(x, y), and take the point M(x + Ax, 
y -f Ay). On letting V denote the volume 
obtained by revolving arc PL about OX, 
the volume obtained by revolving arc LM 
may be denoted by AV. Through L and 
M draw the lines shown in the figure. 

The volume obtained by revolving arc LM 
about the x-axis is greater than the volume 
obtained by revolving LG, and is less than the 
volume obtained by revolving KM. That is, 

v.UL i .LG< AV<tt . VM 2 .KM; 




iry 2 • Ax < A V < rr • (y + Ay) 2 • Ax. 

.-. ?ry 2 < ±K <7r (y + Ay) 2 . 
Ax 



(1) 



On letting Ax approach zero, the three numbers in (1) become 
dV 



Try* 



Hence, 



From (2) and Art. 27 



dx 
dx 



wy 2 , respectively. 



iry'2, 



dV= try* • doc* 



If PQ had been revolved about the ?/-axis, then 
dV 



dy 



= ira?2, and dV = ira?2 • dy. 



(2) 

(3) 



(4) 



Note. According to (3), for a given differential dx the corresponding 
differential of the volume is the volume of a cylinder of radius y and height 
dx. The smaller dx is, the more nearly does this volume become equal to the 
actual increment, due to dx, in the volume of the solid of revolution. 

Ex. 13. Derive the results in (4). 

Ex. 14. (1) Find the x-derivative of the volume generated by the revolu- 
tion of the parabola y = x 2 about the x-axis. (2) Find the ^/-derivative of 
the volume generated by the revolution of this curve about the y-axis. 

Ex. 15. (1) Calculate the differential of the volume in Ex. 14 (1), taking 
dx = .l at the point where x = 2. (2) Thus also in Ex. 14 (2), taking 
dy = .2 at the point where x = 4. (The actual increment in the volume of 
the solid due to changes dx and dy can be computed by Art. 182.) 



67.] GEOMETRIC DERIVATIVES. 101 

(/) Derivative and differential of the area of a surface of revolu- 
tion. Let PQ be an arc of the curve y =/(x). On PQ take any point, say 
L(x, y), and take the point M(x + Ax, y + Ay). Let S denote the area of 
the surface generated by revolving arc PL about OX; then the area generated 
by revolving arc LM about OX may be de- 
noted by AS. There is evidently a straight Y 
line whose length is equal to the length of the 
arc LM. Through L and M draw the lines 
LM' and ML' parallel to OX and equal in 
length to the arc LM. {LM may be supposed 
to be a piece of wire, LM' the same piece of 
wire when it is stretched out in a horizontal 




straight line from L, and ML' the same piece F 24 

of wire when it is stretched out in a horizontal 

line from M. ) The surface obtained by revolving the arc LM about OX is 
greater than the surface obtained by revolving LM' ; for, with the exception 
of the point L, each point on LM has a greater ordinate than the corre- 
sponding point in the line LM', and consequently a greater radius of swing. 
Similarly, the surface obtained by revolving LM is less than the surface 
obtained by revolving ML'. That is, 

2 Try • LM' < surface generated by LM < 2 w {y + Ay) • L'M; 

i.e. . 2 iry . arc LM< AS < 2 tt (y + Ay) • arc LM. (1) 

... 2T y?^M < ^ <2v( y + Ay) mm. (2) 

Ax Ax Ax 

On letting Ax approach zero, the three numbers in (2), by Arts. 20, 22, 
23, 67c, take the values 

• 2 Try—, — , 2 th/—, respectively; 
dx dx dx 

and hence ^? = 2tt?/^. (3) 

dx dx 

On dividing the members in (1) by Ay, and letting Ay approach zero, 

^ = 2ttij^-. (4) 

dy dy 

Similarly, if arc PQ revolve about the ?/-axis, 

^ = 2ttx^ (5), and ^ = 2ttx^. (6) 

dx dx dy dy 

From (3), (4), and Art. 67 (c) [(2), (3)], 



f= 2 ^ + (I)*g=^ + (f)- m 



102 DIFFERENTIAL CALCULUS. [Ch. V. 

Similarly, in case of revolution about the y-axis, from (5) and (6), 



Results (3), (4), (7), show that, for a curve revolving about the a>axis, 
dS = 2 «y • ds = 2 wjry/l + ( J|) 2 to = 2 iry^l + I^Y dy ; (9) 
and (5), (6), (8), show that, for a curve revolving about the y-axis, 

d S = 2 ira; • ds = 2 ir W 1+ (|^) 2 «*» = 2 «» V 1 + (ff )* d5y ' (10) 

Ex. 16. Derive results (5), (6), (8), and (10). 

Ex. 17. Find the ^-derivative and the ^/-derivative of each of the surfaces 
described in Ex. 14. 

Ex. 18. Calculate the differentials of the surfaces described in Ex. 15. 
Make figures showing these differentials. (The actual increments of the 
surfaces can be computed by Art. 211.) 

Ex. 19. Find — , — , — , ^, for the ellipse VW + «V = a 2 b 2 . For 
dx dx dx dx 

a given differential of x, draw figures showing the corresponding differentials 

of s, A, V, and x. 

dt 

Ex.20. Find — for r 2 =a 2 cos20, r=<zcos0, r=ae^ cot «, r=a(l + cos0). 
dd 

Ex. 21. If denote the eccentric angle of the ellipse in Ex. 19, show that 

ds 



— = avl - e' 2 cos 2 0, e being the eccentricity. 
d<f> 



CHAPTER VI. 

SUCCESSIVE DIFFERENTIATION. 

N.B. Article 68 contains all that the beginner will find necessary concern- 
ing successive differentiation for the larger part of the remaining chapters. 
Accordingly, the reading of Arts. 69-72 may be deferred until later. 

68. Successive derivatives. As observed in many of the pre- 
ceding examples, the derivative of a function of x is, in general, 
also a function of x. This derivative, which may be called the 
first derived function, or the first derivative (of the function), may 
itself be differentiated ; the result is accordingly called the second 
derived function, or the second derivative (of the original function). 
If the second derivative is differentiated, the result is called the 
third derived function, or the third derivative ; and so on. If the 
operation of differentiation is performed on a function n times in 
succession, the final result is called the nth derived function, or 
the «th derivative, of the function. 

Ex. If the function is x 4 , then its first derivative is 4x 3 ; its second 
derivative is 12 x 2 ; its third derivative is 24 x ; its fourth derivative is 24 ; 
its fifth and its succeeding derivatives are all zero. 

Notation, (a) If y denote the function of x, then 

the first derivative, namely — (y), is denoted by — (Art. 23) ; 

ax ax 

the second derivative, namely — ( — \ is denoted by — \\ 

dx\dxj dx 2 



the third derivative, namely — 

dx 



"A(%Y1 ^ denoted by % 

dx\dxj] J dx*' 



and so on. On this plan of writing, 



the nth derivative is denoted by ^-^, 

103 



104 DIFFERENTIAL CALCULUS. [Ch. VI. 

In this notation the integers 2, 3, •••, n, are not exponents; 
these integers merely indicate the number of times that the 
function y is to be differentiated successively with respect to x. 

(b) The letter D is frequently used to denote both the opera- 
tion and the result of the operation indicated by the symbol 

(See Art. 23.) The successive derivatives of y are then 

dx y [ u 

Dy, D(Dy), D\D(I)y)'], •••; these are respectively denoted by 

Dy, D*y, &y, .», L>y. 

Sometimes there is an indication of the variable with respect 
to which differentiation is performed ; thus 

D x y, DJy, D x % ..., D»y. 

Note. Here n is not an exponent ; D n y does not mean (Dy) n . {E.g. see 
Exs., p. 108.) D"y is called the derivative of the nth order. 

(c) Instead of the symbols shown in (a) and (b), for the succes- 
sive derivatives of y, the following are sometimes used, namely, 

y',y",y'"> -, */ (n) - 

(d) If the function be denoted by <£(#), its first, second, third, •••, 
and nth derivatives (with respect to x) are generally denoted by 

<t>'(x), <£"(x), <£'"(#), •••, <fi (n) (x) or <j> n (x), respectively; 

d d 2 d 3 d n 

also by — <b(x), — -<t>(x). - — -<b(x). ••♦, —d>(x). 

Note 1. In this book notation (a) is most frequently used. The symbol 
D is very convenient, and is especially useful in certain investigations. See 
Byerly's Biff. Cal., Lamb's Calculus, Gibson's Calculus (in particular § 67). 
For an exposition of simple elementary properties of the symbol D also see 
Murray's Differential Equations (edition 1898), Note K, page 208. 

Note 2. Instead of the accent notation in (c), the ' dot '-age notation, 



is sometimes used, particularly in physics and mechanics. 

Note 3. Geometrical meaning of -=-=5 • It has been seen in Arts. 25, 26, 
that -^- , i.e. — (?/), denotes the rate of change of y, the ordinate of the curve, 



68.] SUCCESSIVE DIFFERENTIATION . 105 

compared with the rate of change of the abscissa x ; this may be simply 
denoted as the x-rate of change of the ordinate. Similarly — |, i.e. -5- (-57), 

is the rate of change of the slope — of a curve compared with the rate of 

change of the abscissa x, or, simply, the x-rate of change of the slope. 

On a straight line, for instance, the slope is constant, and hence the x-rate 
of change of the slope is zero. This is also apparent analytically. Tor, if 

y = mx + c is the equation of the line, then -^ = m, and hence — = 0. 

efts 
Note 4. Physical meaning of -^j^' I n Art. 25 it has been seen that 

if s denotes a varying distance along a straight line, — , i.e. — (s), denotes 

dt dt 

/72a (J /(Jo 

the rate of change of this distance, i.e. a velocity. Similarly — , i.e. — ( — 

J J dt 2 ' dt\dt 

denotes the rate of change of this velocity. Rate of change of velocity is 

called acceleration. For instance, if a train is going at the rate of 30 miles 

an hour, and half an hour later is going at the rate of 40 miles an hour, its 

velocity has increased by ' 10 miles an hour ' in half an hour, i. e. as usually 

expressed, its acceleration is 10 miles per hour per half an hour. Again, it 

is known that if s is the distance through which a body falls from rest 

ds d 2 s 

in t seconds, s — \gt 2 . Hence — = gt ; accordingly, — = g. That is, the 

dt dt 2 

acceleration of a falling body is ' g feet per second ' per second. (See 

text-books on Kinematics, Dynamics, and Mechanics, for a discussion on 

acceleration. ) 

EXAMPLES. 

1. Find the second x-derivative of: (i) xtan-ix; (ii) 4x 2 — 9x + 

- — Vx ; (iii) tan x + sec x ; (iv) x x . 
x 

2. Find D x s y, when: (i) y =(x 2 + a 2 ) tan- 1 ^ ; (ii) y = log (sin x). 

a 

giy 1 

3> Eind dx*' when : ® y = sin ~ lx; ( H ) y = r+i&' 

4. Find D x 6 y, when : (i) y = x 4 log x ; (ii) y = e x cos x. 

d 2 v 

5. Find — |, when xy 1 + 3 x + 5 y = 0. 

By Art. 56, ^ = ,/ + 8 . (!) 

J ' dx 2xy ^ 5 v J 



d 2 v 
On differentiation 



(2xy + 5)22/^ -0/ 2 + 3) (22/ + 2x|^ 



dx 2 (2X2/ + 5) 2 



106 DIFFERENTIAL CALCULUS. [Ch. VI. 

On substituting the value of -^ , and reducing, 
dx 

d 2 y 2(y 2 + S)(Sxy 2 + 10y-Sx) 

dx 2 (2 xy + 5) 3 { } 

6. (i) In the ellipse a 2 y 2 + b 2 x 2 = a 2 b 2 calculate D x 2 y. (ii) Given 
y% -\- y = x 2 , find D x 3 y. 
Work of part (i) : 
Equation of ellipse, a 2 y 2 + b 2 x 2 = a 2 b 2 . (1 ) 

On differentiation, 2a 2 w^ + 2 b 2 x = 0. 
dx 

Whence -/ = j-- C2) 

dx a 2 y v y 



On differentiation in (2) , d?y tf 

w ' dx 2 a 2 

On substitution from (2), and reduction, 



whence, by (1), 



x dy \ 
dx 



d 2 y 
dx 2 


b 2 fa 2 y 2 + b 2 x 2 \ _ 
a 2 \ a' 2 y s J ' 


d 2 y 
dx 2 


b 2 a 2 b 2 _ 6 4 
a 2 dhf ~ a 2 y 3 



7. Show that the point Q, |) is on the curve log (x + y) = as — y. Show 

i 

2' 



that at this point ^ = 0, and ^ = i 



c?x dx 2 

8. What are the values of ^ and ^ : (i) at the point (2, 1) on 

dx dx 2 

the ellipse 7 x 2 + 10 y 2 = 38; (ii) at the point (3, 5) on the parabola 
y 2 = 4 x + 13 ? 

9. Calculate — ^ for the cycloid in Art. 43, Ex. 6. Compute it when 



a = 8 and 6 = w 



dx 2 
3" 

x = a(0 — sin 0), y = a(l — cos 0). 

.-. ^ = a(l-cos0), and ^=asin0. 
d0 v Jl dd 



= p*iU^,byArt.35l = ashld =-™± 
\_dd dd J a(l-cos0) 1-cos 



2 sin - cos - 

2 2 

S— = COt; 

2sin 2 - 

2 



SUCCESSIVE DIFFERENTIATION. 107 

±(ay\a_(dy\ m dd [Art26(1)] 

dx\dx) dd\dx) dx WJ 

A(cot-) + — [Art. 36] 
dd\ 2/ dd L J 



cosec 2 - 

l a e l 2 

— - cosec 



2 2 a(l-cos0) 4asiu2 ^ 4asin4 ^, 

2 2 



d 2 ?/ 1 1 



dx 2 32 sin 4 30° 2 

x 4- b cos x. 

tfx 2 

i _a2)^_ x^ = 2j (iii)ify = a 

b sin (log x), x 2 ^ + x^ 



10. Verify the following : (i) if y = a sin x + 6 cos x, -^ + ?/ = 

(ii) if u = (sin- 1 x) 2 , (1 - x 2 ) — - x — = 2 ; (iii) if y = a cos (log x) + 

dx 2 dx 



dx 2 dx^ y 

11. Show that if u = y 2 logy, and y=f(x), — =(2 logy + 3) f^Y 

, 2 dx 2 V^/ 

+ y(21og?+l)||. 

d% 

12. Find — £ in the following cases : ?/=4x 3 + 2x — 3, w = 4x 3 + 4x + 2, 

dx 2 

y = 4 x 3 + 5 x — 4, ?/ = 4 x 3 + ex + k. 

13. Given that — \ — 3 x + 2, find the most general expression for 

-^ ; then find the most general expression for y. 
dx 

14. A curve passes through the point (2, 3) and its slope there is 1; at 
any point on this curve —^- =2x; find its equation and sketch the curve. 

15. At any point on a certain curve — ^= 8; the curve passes through 

dx 2 

the origin and touches the line y = x there ; find its equation and sketch the 
curve. 

16. (1) In the case of simple harmonic motion, Ex. 13 (p. 78), show 
that the speed of Q is changing at a rate which varies as the distance of Q 
from the centre of the circle. (2) What is the acceleration of the velocity 
of the boat in Ex. 18, Art. 37 ? 

17. In Ex. 14 (p. 78), calculate the rate at which Q is changing its speed 
when Q is : (i) at an extremity of the diameter ; (ii) 12 inches from the 
centre ; (iii) 6 inches from the centre ; (iv) at the centre. 



108 DIFFERENTIAL CALCULUS. [Ch. VI. 

18. A body moving vertically has an acceleration or a retardation of 
g feet per second due to gravitation, g being a number whose approximate 
value is 32.2 : find the most general expression for the distance of the moving 
point from a fixed point in its line of motion, after t seconds. Explain the 
physical meaning of the constants that are introduced in the course of 
integration. 

19. A body is projected vertically upwards with an initial velocity of 500 
feet per second : find how long it will continue to rise, and what height it 
will reach, if the resistance of the air be not taken into account. 

20. A rifle ball is fired through a three-inch plank, the resistance of 
which causes an unknown constant retardation of its velocity. Its velocity 
on entering the plank is 1000 feet a second, and on leaving the plank is 
500 feet a second. How long does it take the ball to traverse the plank ? 
(Byerly, Problems in Differential Calculus.) 

69. The nth derivative of some particular functions. In a few 

cases the nth derivative of a function can be found. This is 
done by differentiating the function a few times in succession, 
and thereby being led to see a connection between the successive 
derivatives. 

EXAMPLES. 

1. Let y = x r . 
Then Dy = rx r ~ x ; 

D-y = r(r— l)x>— 2 ; 

D z y =r(r-l)(r- 2)x'- 3 . 
From this it is evident that 

jyny _ r ( r _!)(»._ 2) ... (r- n + l)x r ~ n . 
Show that D n x n = n ! 

2. Find the nth derivative of the following functions : 

(a) e*; (b) a x ; (c) e ax , (d) a hx . 

3. Show that the wth derivative of sin x is sin ( x -\ — - j • 
Suggestion: cos z = sin (z + - )• 

4. Find the nth derivatives of (a) cos x ; (&) sin ax ; (c) cos ax. 

5. Find the nth derivatives of log x, log (x — 2) 2 . 



69-71.] SUCCESSIVE DIFFERENTIATION. 109 

6. Find the nth derivatives of - , 



7. Find the nth derivatives of 



x 1 + x 3 — x (?) + ex)' 
2 2x 



1 _ X 2 1-3-2 

[Suggestion : Take the partial fractions.] 

70. Successive differentials. In Art. 27 it has been shown that if 

V =/(*). (1) 

then dy=f'(x)dx. (2) 

The differential in (2) is, in general, also a function of x ; and its differ- 
ential may be required. In obtaining successive differentials it is usual to 
give a constant differential increment dx to x. Then (Art. 27), on taking 
the differentials of the members in (2), 

d(dy) = d [/'(«)*&] = [f"(x)dx]dx. (3) 

On taking the differentials of the members of (3), 

d{d(dy)} = d{[f"(x)dx~] dx}=f "(x)dx • dx ■ dx. (4) 

It is customary to denote results (3) and (4) thus : 

&y=f'(x)cfo? and dhj =f'"(x)dx\ 

In this notation the nth differential is written 

d n tj =f n (x)dx n , 

in which f n (x) denotes the nth. derivative of /(#), and dx n denotes (dx) n . 

71. The successive derivatives of / with respect to x when both 
are functions of a third variable, t say. 

An example will show the method of finding these derivatives. 

EXAMPLES. 

1. Find ^ and ^, when x = 2 + 5 t - f- (1) 

dx dx 2 

and y - 8 1 - P ; (2) 

'. w] 
dx' df 2 



also find x, y, -X — ^, when t = 2 



From (1), ^=5_2£. (3) 

From (2), ^ = 8 - 3 f 2 . (4) 

f7« 



110 DIFFERENTIAL CALCULUS. [Ch. VI. 



dy 

.%*=£ '(Art 86) =*=**■ 

dx dx y J 5-2? 

dt 



(5) 



,^ = ±(dy\±fdy\ dt_ iATLM)= £/dy\dx (Art< 36) 
dx 2 dx\dx) dt\dx) dx dt\dx) dt 



= KHi)-*rr ( ^(3) and (5)] 
6 £ 2 - 30 « + 16 



(5-2 £) 3 
If t = 2, then by (1), (2), (5), (6), 



(6) 



x = 8, 2, = 8, 4=,-4, f^ 2 =-20. 

2. See Ex. 9, Art. 68. 

3. Find D x y and D x 2 y when x = a — 6 cos and y = a0 + b sin 0. 

4. Find -^ and — ^ in the following cases : 

dx dx? 

1 — t 2t 

(i) x= , w = ; (ii) x = acos0, y = asmd; (iii) x = acos0, 

1 + t 1 + t y 

?/ = 6 sin ; (iv) x = cot t, y = sin 3 1. 

72. Leibnitz's theorem. This theorem gives a formula for the nth 
derivative of the product of two variables. Suppose that u and v are func- 
tions of x, and put y = uv. 

Then, on performing successive differentiations, 

Dy = u •" Dv -f v • Dw ; 

D 2 y = u • D 2 v + 2Du • Dv + v • D 2 u ; 

1% = u • Z>% + 3 Du • 2> 2 i? + 3 Dhi . Dv + v • Dhi ; 

D*y = m • D 4 u + 4 Dm ■ Z>% + 6 Z>% . Z> 2 v + 4 Z)% • Dv + v . 2)%. 

Thus far the numerical coefficients in these derivatives are the same as the 
numerical coefficients in the expansions (a+v), (u + v) 2 , (u + v) 3 , and 
(u + vy respectively, and the orders of the derivatives of u and v are the 
same as the exponents of u and v in those binomial expansions. Now sup- 
pose that these laws (for the numerical coefficients and the orders) hold in 
the case of the nth derivative of uv ; that is, suppose that 

D n (uv) = u ■ D n v + nDu ■ D n ~H + n ( n ~ ^ D 2 u • D n ~ 2 v + — 

1 • 2 

n(n-l)--(n-r + 2) 2>r _ lf< > Dn _ r+h} 
].2...(r-l) 

+ ^ - 1) - (n - r + 1) D r u . j-^ + ... + P . d» m . (1) 
1 .2 ••• r 



72, 73.] SUCCESSIVE DIFFERENTIATION. Ill 

Then these laws for the coefficients and the orders hold in the case of the 
(n + l)th derivative of wo. For differentiation of both members of (1) gives 

D'^ 1 (uv) = u • B*+h>+(n + l)Du • D*o + + 1 )» j)hi ■ D'^v + ••• 

+ (n + l)n(n-l)~.(n-r + 2) ^ . ^_ r+ly + ... + „ . ^^ 
l-2...(r-l)r 

Hence, if formula (1) is true for the nth derivative of uv, a similar formula 
holds for the (n + l)th derivative. But, as shown above, formula v (l) is true 
for the first, second, third, and fourth derivatives of uv ; hence it is true for 
the fifth, and for each succeeding derivative. 

Ex. 1. Find D x n y when y = x-e x . 

D n y = x- • D»(e*) + nD(x z ) ■ D*" 1 ^) + n(jl - ^ D 2 (x 2 ) • D n ~\e x ) + ... 

= e*[> 2 -f 2 nx + n(n- 1)]. 
Ex. 2. Calculate the fourth ^-derivative of x° sin x by Leibnitz's theorem. 
Ex. 3. Eind D x n y when : (i) y = a-e x ; (ii) ?/ = xe 2x . 

Note. Eeference for collateral reading on successive differentiation. 
Echols, Calculus, Chap. IV., especially Art. 56. 

73. Application of differentiation to elimination. It is shown in 
algebra that one quantity can be eliminated between two inde- 
pendent equations, two quantities between three equations, and 
that n quantities can be eliminated between n -f- 1 independent 
equations. The process of differentiation can be applied for the 
elimination of arbitrary constants from a relation involving vari- 
ables and the constants. For by differentiation a sufficient num- 
ber of equations can be obtained between which and the original 
equation the constants can be eliminated. 

EXAMPLES. 

1. Given that y = A cos x + B sin x, (1) 

in which A and B are arbitrary constants, eliminate A and B. 

In order to render possible the elimination of these two constants, two 
more equations are required. These equations can be obtained by differen- 
tiation. Thus, 

^ = — A®nx + B cos jc, (2) 

dx 

^4 = — A cos x — B sin x. (3) 

dx 2 



112 DIFFERENTIAL CALCULUS. [Ch. VI. 

On eliminating A and B between (1), (2), (3), there is obtained the relation 

Note 1. Equation (4) is called a differential equation, as it involves a 
derivative. It is the differential equation corresponding to, or expressing 
the same relation as, the " integral" equation (1). The process of deducing 
the integral equations (or solutions, as they are then called) of differential 
equations is discussed, but for a very few cases only, in Chapter XXVII. 

2. Eliminate the arbitrary constants m and b from the equation 

y = mx + b. Ans. — ^ = 0. 

dx 2 

In this case the given equation represents all lines, m and b being arbi- 
trary. Accordingly the resulting equation is the differential equation of all 
lines. For the geometrical point of view see Art. 68, Note 3. 

3. Eliminate the arbitrary constants a and b from each of the following 
equations : (1) y = ax 2 + b. (2) y = ax 2 + bx. (3) (y — b) 2 = 4 ax. 

(4) y 2 - 2 ay + x 2 = a 2 . (5) y 2 = 6 (a 2 - x 2 ). 

4. Find the differential equations which have the following equations for 
solutions, Ci and c 2 being arbitrary constants : 

(1) y = d. (2) y = ax. (3) y = c 1 x+ c 2 . (4) y = c x e x + c 2 e~ x . 

(5) y=cie mx +c 2 e- mx . (6) y=Ci cos wise + c 2 sinmx. (7) y=C\ cos (mx +c 2 ). 

5. Obtain the differential equations of all circles of radius r: (1) which 
have their centres on the cc-axis ; (2) which have their centres on the ?/-axis ; 
(3) which have their centres anywhere in the £#-plane. 

6. Show that the elimination of n arbitrary constants ci, c 2 , •••, c„, from 
an equation /(cc, y, c±, c 2 , •••, c M ) = gives rise to a differential equation 
involving the nth. derivative of y with respect to x. 

Note 2. For geometrical explanations relating to differential equations 
the student is referred to Murray, Differential Equations, Chap. I., which 
may easily be read now. The reading will widen his mathematical outlook 
at this stage. 



CHAPTER VII. 

FURTHER ANALYTICAL AND GEOMETRICAL 
APPLICATIONS. 

VARIATION OF FUNCTIONS. SKETCHING OF GEAPHS. 
MAXIMA AND MINIMA. POINTS OF INFLEXION. 

tf.B. This chapter may be studied before Chapter V. is entered 
upon. 

74. Increasing and decreasing functions. When x changes con- 
tinuously from one value to another, any continuous function of x, 
say cf>(x), in general also changes. The function may either be 
increasing or decreasing, or alternately increasing and decreas- 
ing. By means of the calculus it is easy to find out how the 
function behaves when x passes through any value on its way 
from — x to + x . 

Let Ax be a positive increment of x, and A</>(V) be the corre- 
sponding increment of <£(.r). If <£(V) continually increases when x 
is changing from x to x + Ax, then A<f>(x) is positive ; and accord- 
ingly, ^' ' is positive. Moreover, this is positive for all posi- 
Ax 

tive values of Ax, however small; hence lim Azi0 — £i£Z ? i, e . <f>'(x), is 

... Ax 

positive or zero. 

Similarly, if <f>(x) continually decreases when x is increasing 

from x to x + A.r, <f>'(x) is negative or zero. In other words : 

If <p(x) is increasing in an interval, <p'(x) is positive or zero for values 
of x in the interval ; 

if <f>(x) is decreasing in an interval, <j>'(x) is negative or zero for values 
of x in the interval. 

On the other hand : 

If <t>'(x) is always positive in an interval, <j>(x) is constantly increas- 
ing in the interval ; 

if <p'{x) is always negative in an interval, <f>(x) is constantly decreas- 
ing in the interval. 

113 



A. 



114 



DIFFERENTIAL CALCULUS. 



[Ch. VII. 



The case when 4>'(x) is zero will be discussed later. 

Properties A and B are illustrated by Figs. 25 a, b, c ; 26 a, b, 
c, d, e, f. 

Let 4>(x) be graphically represented by the curve ABCDE, 
whose equation is 

V = <£(»• 

At any point on this curve, -^= <f)'(x). 

dx 

By Art. 24, the slope of the curve represents the derivative of 
the function. Now at A, D, and E, the slope is negative, and the 
ordinate y (the function) is evidently decreasing as x is passing in 
the positive direction through the values of x at A, D, and E. 
On the other hand, at B, C, and F, the slope is positive, and the 
ordinate y is evidently increasing as x is passing in the positive 




Y 












\ 


L 


N 






\ 


M 








Li 


Mi 


Nx 


O 


< — i 


n — > 




X 



Y 


J 


c 






J 


V 


^TF 





< ft " 




X 



Fig. 25 6. 



Fig. 25 c. 



Fig. 25 a. 



direction through the values of x at B, C, and F. In Fig. 25 5 
when x is increasing from OL x to OJ^, the ordinate y is decreas- 
ing from L^L to i^TWand the slope at points on LM is negative; 
when x is increasing from OM 1 to OA^, the ordinate is increasing 
from M X M to N Y N and the slope at points on MN is positive. 
Fig. 26 a shows functions increasing or decreasing in an inter- 
val which have a zero derivative within the interval. 



75. Maximum and minimum values of a function. Critical points 
on the graph, and critical values of the variable. The values of the 
function at points such as P lt P 2 , P 3 , M, and IT (Art. 74), where 
the function stops increasing and begins to decrease, or vice versa, 



75.] 



MAXIMUM AND MINIMUM. 



115 



may be called turning values of the function. When a function 
ceases to increase and begins to decrease, as at P 2 , P 4 , and K, it is 
said to have a maximum value ; when a function ceases to decrease 
and begins to increase, as at P D P 3 , and M, it is said to have a 
minimum value. Therefore, at a point (on the graph) where the 
function has a maximum value the slope changes from positive to 
negative ; at a point where the function has a minimum value the 
slope changes from negative to positive. (Examine Fig. 25.) 
Accordingly, at each of these points the slope (i.e. the derivative of 
the function) is generally (see Note 3) either zero or infinitely 
great. 

It should be observed that, although the derivative of a function 
may be either zero or infinitely great for values of the variable for 
which the function has a maximum or a minimum value, yet the 
converse is not always the case. The function may not have a 
maximum or minimum value when its derivative is zero or infinity. 




V 




o 

Fig. 26 &. 



This is exemplified by the functions whose graphs are given in 
Figs. 26 a, b. Thus at P the slope is zero and the function is 
increasing on each side of P; at Q the slope is zero and the 
function is decreasing on each side of Q ; at R the slope is infi- 
nitely great, and the function is increasing on each side of R ; 
at S the slope is infinitely great and the function is decreasing 
on each side of S. 

Accordingly, a point where the slope of a graph of a function 
is zero or infinitely great is, for the purpose of this chapter, called 
a critical point. Such a point must be further criticised, or ex- 
amined, in order to determine whether the ordinate has either a 
maximum or a minimum value there. In other words, that value 



116 DIFFERENTIAL CALCULUS. [Ch. VII. 

of the variable for which, the derivative of a function is zero or 
infinitely great is called a critical value; further examination is 
necessary in order to determine whether the function is a maxi- 
mum or a minimum for that value of the variable. 

Note 1. The points Q, P, B, S (Figs. 26 a, 6), are examples of what are 
called poinds of inflexion (see Art. 78). 

Note 2. By saying that a function 0(x) has a minimum value, for x = a 
say, it is not meant that 0(a) is the least possible value the function can 
have. It is meant that the value of the function for x = a is less than the 
values of the function for values of x which are on opposite sides of a, 
and as close as one pleases to a ; i.e. h being taken as small as one pleases, 
0(a) < 0(a — h) and 0(a) < 0(a + h). (See Pi in Fig. 25 a.) Likewise, if 
0(x) is a maximum for x = 6, this means merely that 0(6) >0(6 — h) and 
<t>(b) > 0(& + h),in which h is as small as one pleases. (See P 2 in Fig. 25 a.) 



EXAMPLES. 

1. Examine sin x for critical values of the variable. 

Here 0(x)=sinx. 

The graph of this function is on page 459. In order to find the critical 
points solve the equation 

0'(x)= cosx = 0. 
Accordingly, the critical values of x are -, '— - , — , ••-. 

2. Examine (x — l) 2 (x + 3) for critical values of the 
variable. 




Here 0(z) = (x - l) 2 (x + 3). 

The solution of 0' (x) = (x - 1) (3 x + 5) = 0, 



3. Examine (x — l) 3 + 2 for critical values of the 
variable. 

Here 0(x) =(x - 1) 3 + 2. 

On solving <f>'(x) = 3(x - l) 2 = 0, 

Fig. 26 d. the critical value of x is obtained, viz. x = 1. 



re.] 



MAXIMUM AND MINIMUM. 



117 



4. Examine (x — 2) 3 + 3 for critical values of x. 



Here 

On solving 



0(x)=(x-2)3+3. 



3(:k -2)3 
the critical value x = 2 is obtained. 

5. Examine (x — 2) ¥ + 3 for critical values of x. 

i 



Here 
and 



0Ob) = (x-2)3+3 
0'(b) = - r = oo 



3(3 - 2)* 

gives the critical value x = 2- 




Fig. 26 /. 



Note 3. A function may have a maximum or minimum value when its 
derivative changes abruptly ; see Art. 164, Note 3, and Fig. 21 (c), Infin. Cal. 



76. Inspection of the critical values of the variable (or critical 
points of the graph) for maximum or minimum values of the function. 
Let the function be <f>(x). The equation of its graph, is y = <f>(x), 

and the slope is -2 or <j>'(x). The solutions of the equations 
dx 

<fi'{x) = and <f>'(x) = go , 

give the critical values of the variable. 

Suppose that ABCDE (Fig. 25 a) is the graph, and that the 
critical values are x = a and x=b. There are three ways of 
testing whether the critical values of the variable will give maxi- 
mum or minimum values of the function, viz. : 

(a) By examining the function itself at, and on each side of, 
the critical value ; 

(b) By examining the first derivative on each side of the 
critical value ; 

(c) By examining the second derivative (see Art. 68) at the 
critical value. 

Note 1. It follows from the definition of maximum and minimum values, 
and Note 2, Art. 75, that if 0(a) is a maximum (or minimum) value of 0(x), 
then 0(a) + ra, c<p(a), v'0(«)' 2 («)i •••, are maximum (or minimum) 



118 DIFFERENTIAL CALCULUS. [Ch. VII. 

values of <f>(x)+m, c0(x), V<p(x), 2 (x), •••, respectively. Accordingly, 
the finding of critical values of x for one of these functions will give the 
critical values for the other functions. It sometimes happens that it is much 
easier to find the critical values for, say 2 (x), than for 0(x). In such a 
case it is better to examine <p 2 (x) than to examine <p(x). 

(a) Examination of the function. Let <£(#) denote the function, 

and x = a be the critical value of x. 

In this test the value of <j>(a) is compared with two values of 
<j>(x), viz. when x is a little less than a, and when a? is a little 
greater than a; say, when x — a — h and when x = a + A, in which 
h is a small number. 

If <f>(a) is greater than both <j>(a — h) and <f>(a + h), <f>(a) is a maxi- 
mum (as at P 2 1EL Fig. 25 a and Km Fig. 25 c). 

Tjf <j>(a) is less than both <f>(a — h) and <ft(a + h), <£(a) is a minimum 
(as at Pj and P 3 in Fig. 25 a and Min Fig. 25 6). 

If <f>(a) is greater than the one and less than the other of <f>(a—h) 
and <f>(a-\-K) t <f>(a) is neither a maximum nor a minimum (as 
at P, Q, E, S, Figs. 26 a, b, and at x = 1 in Fig. 26 d). 

Ex. 1. In Ex. 1, Art. 75, examine the function at the critical value - of x. 

Here sin ( * _ ft J < sin - , and sin j ^ + ft ) < sin - • Accordingly, x=- 
\ 2 / 2 \'2i ) A 2 

gives a maximum value of sin x. 

Ex. 2. (a) In Ex. 2, Art. 75, examine the function at the critical value 
x=l. Here 0(1) =0, 0(1 - ft) = A»(4-fc), 0(1 + ft)= ft2(4 + ft). Accord- 
ingly, 0(1 — A) > 0(1), and 0(1 + ft)>0(l). Thus 0(1) is a minimum 
value of 0(as). 

(&) Inspect this function at the critical value x =— f. 

Ex. 3. In Ex. 3, Art. 75, examine the function at the critical value x = 1. 
Here 0(1) = 2, 0(1 - ft) = - ft 3 + 2, and 0(1 + ft) = ft 3 + 2. Accordingly, 
0(1 — ft) < 0(1) < 0(1 + ft), and thus 0(1) is not a turning value of the 
function. 

Ex. 4. Examine the functions in Exs. 4, 5, Art. 75, at the critical 
values of x. 



76.] MAXIMUM AND MINIMUM. 119 

(6) Examination of the first derivative of the function. When 
the derivative of a function is positive, the slope of its graph is 
positive and the function is increasing; when the derivative is 
negative, the slope of the graph is negative and the function is 
decreasing (Art. 74). Hence, h being taken as small as one 
pleases, if <f>' (a — h) is positive and <f>'(a-\-h) is negative, then <j>(a) 
is a maximum value of <f>(x). On the other hand, if <f}'(a — h) is 
negative and <f>'(a + h) is positive, then <j>(x) is decreasing when x 
is approaching a, and <f>(x) is increasing when x is leaving a, and 
accordingly <f>(a) is a minimum value of <j>(x). Examine Figs. 25 
at, and near, P i} P 2 , P 3 , M } K. 

Note 2. Test (6) is generally easier to apply than test (a). For test (a) 
the functions <p(a — h) and 0(a + h) must be computed ; for test (6) merely 
the algebraic signs of <j>'(ja — h) and 0'(« + K) are required. 



Ex. 5. (a) In Ex. 1, Art. 75. 0' h ] is positive and 0' - + h ] is nega- 
tive. Accordingly, 0( -], i.e. sin- or 1, is a maximum value of sinx. 
(6) Apply this test at the other critical values in Ex. 1, Art. 75. 

Ex. 6. (a) In Ex. 2, Art. 75, 0'(1 — h) is negative and 0'(1 + h) is posi- 
tive. Accordingly 0(1), i.e. 0, is a minimum value of (x — l) 2 (x + 3). 

(&) Apply this test at the other critical value in Ex. 2, Art. 75. 

Ex. 7. In Ex. 3, Art. 75, 0'(1 — Ji) is positive and 0'(1 + h) is positive. 
Accordingly, 0(1), or 2, is neither a maximum nor a minimum. 

Ex. 8. Apply test (6) at the critical values of the functions in Exs. 4, 5, 
Art. 75. 

(c) Examination of the second derivative of the function. It has 

been seen that the sign of the derivative of a function <f>(x) changes 
from positive to negative when the function is passing through a 
maximum value. If the derivative <f>'(x) passes from a positive 
value to zero, and then becomes negative, the derivative is contin- 
ually decreasing, and hence its derivative, namely <£"(#), must be 
= , or <, for the critical value of x. On the other hand, when 
the function passes through a minimum value, the derivative 



120 DIFFERENTIAL CALCULUS. [Ch. VII. 

changes sign from negative to positive. If then the derivative 
<f>'(x) passes through zero, it is continually increasing, and hence 
its derivative, namely <£"(#), must be =, or >, for the critical 
value of x. Therefore, 

if <f>'(a) is zero and </>"(a) is negative, <£(a) is a maximum value 
of<j>(x); 

if $'(a) is zero and <}>"(a) is positive, <£(a) is a minimum value 
of<j>(x). 

Note 3. When <f>"(a) is zero, one of the other tests can be used. 
Another procedure that can be adopted when 0"(a) = 0, is discussed in 
Art. 155. 

Note 4. When the second derivative can be obtained readily, test (c) is 
the easiest of the three tests to apply. 

Note 5. Historical. Kepler (1571-1630), the great astronomer, "was 
the first to observe that the increment of a variable — the ordinate of a curve, 
for example — is evanescent for values infinitely near a maximum or minimum 
value of the variable. 1 ' Pierre de Fermat (1601-1665), a celebrated French 
mathematician, in 1629 found the values of the variable that make an ex- 
pression a maximum or a minimum by a method which was practically the 
calculus method (Art. 75). 

Note 6. Many problems in maxima and minima may be solved by ele- 
mentary algebra and trigonometry. For the algebraic treatment see 
(among other works) Chrystal, Algebra, Part II., Chap. XXIV. ; William- 
son, Diff. Cal., Arts. 133-137 ; Gibson, Calculus, § 76 ; Lamb, Calculus, 
Art. 52. 

Note 7. Maxima and minima of functions of two or more inde- 
pendent variables. For discussions of this topic see McMahon and Snyder, 
Diff. Cal, Chap. X., pages 183-197; Lamb, Calculus, pages 135, 596-598; 
Gibson, Calculus, §§ 159, 160 ; Echols, Calculus, Chap. XXX. ; and the 
treatises of Todhunter and Williamson. 



EXAMPLES. 

9. (a) In Ex. 1, Art. 75, tf>" (x) = — sin x. Accordingly, 0"(^) 
negative, and thus 0(-), i-e- sin -, is a maximum value of </>(x). 
(6) Apply test (c) at the other critical values of sin x. 



77.] PROBLEMS IN MAXIMA AND MINIMA. 121 

10. («) In Ex. 2, Art. 75, 0"(sc) = 2(3x -f 1). Accordingly, 0"(1) is 
positive, and thus 0(1) is a minimum value of 0(x). 

(6) Apply test (c) at the other critical value in Ex. 2, Art. 75. 

11. In Ex. 3, Art. 75, <p"(x)= Q(x - 1). Here 0"(1)=O, and thus 
test (c) fails to indicate whether 0(1) is a turning value of 0(x). (See Note 4.) 

12. Apply test (c) at the critical values of the functions in Exs. 4, 5, 
Art. 75. 

Note 8. Sketching" of graphs. The ideas discussed in Arts. 74-76 are a 
great aid in making graphs of functions, and in showing what is termed the 
march of a function. 

13. For each of the following functions find the critical values of x, 
determine the maximum and minimum values, and sketch the graphs : 
(1) 2 x 3 + 5 x 2 - 4 x + 2 ; (2) 5 + 12 x - x 2 - 2 x 3 ; (3) x 2 (x + 1) (x - 2) 3 ; 

(4) (x-2) 3 (x + l) 2 ; (5)2 + 3(x-4)f+(x-4)t; (6) 3 x 5 -125 x 3 + 2160 x ; 
(7) x 2 -7x + 6 , (g) ^z|2f ; (9)xlogx;(10)x*;(ll)2sin 2 x + 8cos 2 x; 
(12) sin # sin 2 x; (13) x cos x. 

14. Show that a + (x — c) n is a minimum when x = c, if n is even ; 
and that it has neither a maximum nor a minimum value, if n is odd. 

15. (a) Show that (4 ac — b 2 ) -4- 4 a is a maximum or a minimum value 
of ax 2 + &x + c, according as a is positive or negative. (6) Show that 
ax 2 + bx + c cannot have both a maximum and minimum value for any 
values of a, &, c. 

16. Find the point of maximum on the curve x z + y 3 — 3 accy = 0. 
Sketch the graph, taking a = 1. 

17. In the case of the ellipse ax 2 + 2 ftx?/ + by 2 + c = 0, show how to 
find the highest and lowest points, and the points at the extreme right and 
left. 

77. Practical problems in maxima and minima. Some practical 
applications of the principles of Arts. 75 and 76 will now be 
given. In making these applications the student is in a position 
analogous to his position in algebra when he applied his knowledge 
about the solution of equations to solving "word problems." Here, 
as in algebra, the most difficult part of the work is the mathe- 
matical statement of the problem and the preparation of the data 
for the application of the processes of Art. 76. 



122 



DIFFERENTIAL CALCULUS. 



[Ch. VII. 



EXAMPLES. 



1. Find the area of the largest rectangle that can be inserted in a 
given triangle, when a side of the rectangle lies on a side of the triangle. 

Let ABC be the given triangle, and let 
the given values of the base AB and the 
height CD be b and h respectively. 

Suppose that M Q is the largest rectangle, 

and let MN and NQ be denoted by y and x 

respectively, and denote the area of M Q by u. 

Then u = xy, which is to be a maximum. 

It is first necessary to express u, the 

quantity to be "maximised," in terms of a 

Fig. 27. single variable. 



M 






H\ 


.P 


h 


r N 


f 


D 


\ 










B 




« 


b 


— > 





Now 



du 



MP : AB = CH : CD ; i.e. x :b = h - y :h. 
.\ x — - (h — y) ; accordingly, u — - y(h — y) , a maximum. 



(h — 2y) = ; whence y 



dy h 
MQ = \bh = one half the area of the triangle. 



Thus x = lb, and area 



Note 1. If M be supposed to move along A C from A to C, the rectangle 
ilf $ increases from zero at A and finally decreases to zero at C. It is thus 
evident that for some point between A and C the rectangle has a maximum 
value. 

Note 2. In these examples it is necessary that the quantity to be maxi- 
mised or minimised be expressed in terms of one variable. Conditions 
sufficient for this must be provided. 

2. Solve Ex. 1, expressing u in terms of x. 

3. A parabola y 2 = Sx is revolved about the #-axis ; find the volume 
of the largest cylinder that can be inscribed in the 

paraboloid thus generated, the height of the parab- 
oloid being 4 units. 

Let OPL be the arc that revolves, LN be at 
right angles to OX, and OV = 4. Take P(se, y), 
a point in OL, and construct the rectangle PV. 
When OPL generates the paraboloid, PV gen- 
erates a cylinder. (As P moves along the curve 
from to X, the cylinder increases from zero at 
and finally decreases to zero at L. Thus there 
is evidently some position of P between and L 
for which the cylinder is a maximum.) Suppose Fig. 28. 



r 


V, 


J 


L^ 






I 

G 


r 


N 





< x > 




X 






u 







77.] 



EXAMPLES. 



123 



that PJY generates the maximum cylinder, and denote its volume by V. 
V = ttPG 2 • GN = iry*(4 - x) = 8 ttz(4 - x). 
Accordingly, — = 8 tt(4 - 2 x) = 0. 



From this, 



x = 2 ; hence F= 100.53 cubic units. 




Fig. 29. 



Note 3. In the process of maximising in Exs. 1, 2, the constant factors ° 
and 8 ir may as well be dropped. (See Art. 76, Note 1.) 

Note 4. In each of these examples it is well to perceive at the outset that 
a maximum or a minimum exists. 

4. A man in a boat 6 miles from shore wishes 
to reach a village that is 14 miles distant along 
the shore from the point nearest to him. He can 
walk 4 miles an hour and row 3 miles an hour. 
Where should he land in order to reach the village 
in the shortest possible time ? Calculate this 
time. Let L be the position of the boat, M the 
village, and N the nearest land to L. Then LN 
is at right angles to NM. Let P denote the place 
to land, and T denote the time (in hours) to go 
over LP + Pdf, and denote NP by x. 



Then 



Hence, 

5. What must be the ratio of the height of a Norman window of given 
perimeter to the width in order that the greatest possible amount of light may 

be admitted ? (A Norman window consists 
of a rectangle surmounted by a semicircle.) 

Let m denote the given perimeter, 2 x the 
width, and y the height of the rectangle in the 
window desired ; let A denote the area of 
the window. 

Then A = 2 xy + £ ttx 2 . 

Now 2x + 2y + Trx = m. 

.-. A = mx — 2 x 2 — | ttx 2 , 

which is to be a maximum. 

On finding the value of x for which A is a 
maximum, and then getting the corresponding value of y, it will appear that 
X = y. Accordingly, the height MB = the width AB. 



T 


_LP PM_ 

3 4" 


V36 + x 2 
3 


14 -x 
4 ' 


a minimum. 


dT 


X 


4 












dx 


<V36 + x 2 




X 


= 6.8 miles, and r = 4.8» 


hours. 





D 





E 






C 


I 






M 


A 




B 









Fig. 30. 



124 DIFFERENTIAL CALCULUS. [Ch. VII. 

6. Find the area of the largest rectangle that can be inscribed in an 
ellipse. (First show that there evidently is such a rectangle.) 

Suggestions : Let the semiaxes of the ellipse be a and 6, and choose 
axes of coordinates coincident with the axes of the ellipse. Let P(x, y) be a 

vertex of the rectangle. Then area rectangle = 4 xy = 4 - xVa 2 — x 2 . Maxi- 

a 

mise the last expression, or, better still, because it is easier to do, maximise 

the square of xVa 2 — x 2 , viz. x 2 (a 2 — x 2 ). (See Art. 76, Note 1.) It will be 

found that the area of the rectangle is 2 a&, half the area of the rectangle 

circumscribing the ellipse. 

7. Divide a number into two factors such that the sum of their squares 
shall be as small as possible. 

8. Two sides of a triangle are given : find, by the calculus, the angle 
between them such that the area shall be as great as possible. 

9. Find the largest rectangle that can be inscribed in a given circle. 

10. Through a given point P(a, 5) a line is drawn meeting the axes 
in A and B ; is the origin : Find (i) the least length that AB can have ; 
(ii) the least value of A + OB ; (iii) the least possible area of the triangle 
OAB. 

11. A and B are points on the same side of a straight line MN: 
determine the position of a point C in MN: (1) so that AC 2 + CB" = a 
minimum ; (2) so that AC + CB = a minimum. 

Jf.B. The cones and cylinders in the following examples are right circular : 

12. (i) Find the height of the cone of greatest volume that can be in- 
scribed in a sphere of radius r. (ii) Find the cone of greatest convex surface 
that can be inscribed in this sphere. 

. 13. Find the semi-vertical angle of the cone of least volume that can be 
described about a sphere. 

14. (i) Find the cylinder of greatest volume that can be inscribed in a 
sphere of radius r. (ii) Find the cylinder of greatest curved surface that 
can be inscribed in this sphere. 

15. (i) Determine the maximum cylinder that can be inscribed in a 
right circular cone of height b and radius of base a. (ii) Determine the 
cylinder of greatest convex surface that can be inscribed in this cone. 

16. What is the ratio of the height to the radius of an open cylindrical 
can of given volume, when its surface is a minimum ? 

17. A circular sector of given perimeter has the greatest area possible: 
find the angle of the sector. 

18. It is required to construct from two circular iron plates of radius 
a a buoy, composed of two equal cones having a common base, which shall 
have the greatest possible volume : find the radius of the base. 



78.] 



POINTS OF INFLEXION. 



125 



19. An open tank of assigned volume has a square base and vertical 
sides : if the inner surface is the least possible, what is the ratio of the depth 
to the width ? 

20. From a given circular sheet of metal it is required to cut out a 
sector so that the remainder can be formed into a conical vessel of maximum 
capacity : show that the angle of the sector removed must be about 66°. 

21. In a submarine telegraph cable the speed of signalling varies as 

x 2 log -, where x is the ratio of the radius of the core to that of the covering : 

x 
show that the speed is greatest when the radius of the covering is Ve times 
the radius of the core. 

22. Assuming that the power required to propel a steamer through still 
water varies as the cube of the speed, find the most economical rate of 
steaming against a current which is running at a given rate. 

23. Assuming that the strength of a rectangular beam varies as the 
product of the breadth and the square of the depth of its cross-section, find 
the breadth and depth of the strongest rectangular beam that can be cut from 
a cylindrical log, the diameter of whose cross-section is d inches. 

24. Find the length of the shortest beam that can be used to brace a 
vertical wall, if the beam must pass over another wall that is a feet high and 
distant b feet from the first wall. 

25. At what distance above the centre of a circle of radius a must an 
electric light be placed in order that the brightness at the circumference of 
the circle may be the greatest possible ? (Assume that the brightness of a 
small surface A varies inversely as the square of the distance r from a source 
of light, and directly as the cosine of the angle between r and the normal to 
the surface at A.) (Gibson's Calculus.) 

78. Points of inflexion : rectangular coordinates. As a point 
moves along the curve LAM from L to M, the tangent at the 
moving point changes from the position shown at L to that at A 

Y 
M 






Fig. 31 6. 



and then to that at M. In going from the position at L to the 
position at A, the tangent turns in the direction opposite to that 
in which the hands of a watch revolve ; in going from the position 



126 DIFFERENTIAL CALCULUS. [Ch. VII. 

at A to the position of M, the tangent turns in the same direction 
as that in which the hands of a watch revolve. Points such as 
A, D, H, G (Fig. 31), and Q, P, R, S (Figs. 26 a, b), at which the 
tangent for the point moving along the curve ceases to turn in 
one direction and begins to turn in the opposite direction, are 
called points of inflexion. 

Examination of the curve for points of inflexion. As the moving 

point goes along the curve from L to A, — increases and accord- 

■j2 ax 

ingly its derivative — ^ is positive ; as the moving point goes 

y-J 72 

along the curve from A to M, — decreases, and accordingly — a 

dx d , dx 2 

is negative. Thus in the case of the curve LAM, —^- is positive on 

-. dx 2 

one side of A and negative on the other. Now -^ changes continu- 

d 2 v ^ X 
ously from L to M : accordingly, at A — 4 = 0. Hence, in order 

dx 2 

to find the points of inflection for a curve y = f(x), proceed as 
follows : 

Calculate ~r-» '-> 

d 2 v 
then solve the equation — % — 0. 

dx 2 

This will give critical values (or points) which are to be further 
examined or tested. A critical point is tested by finding whether 

d 2 v • d 2 v 

— ^ has opposite signs on each side of the point. If — " 2 has oppo- 

(J-y 

site signs, the critical point is a point of inflexion; if —*- has the same 

ax 

^^ ^ ^- sign on both sides of the critical 

-^ "^ point, as in Fig. 31 c, the point is 

what is called a point of undulation. 

Note 1. At a point of inflexion the tangent crosses the curve. The tan- 
gent at an ordinary point on a curve is the limiting position of a secant when 
two of the points of intersection of the 
secant and the curve become coincident 
(Art. 24). The tangent at a point of in- 
flexion is the limiting position of a secant 
which cuts the curve in more than two 
points, when the secant revolves until three 
points of intersection become coincident. 




78.] EXAMPLES. 127 

Thus PT, the tangent at the point of inflexion P, is the limiting position 
of the secant MPQ when JIPQ revolves about P until M and Q simultane- 
ously coincide with P. At a point of undulation the tangent does not cross 
the curve. The tangent at a point of inflexion is called an inflectional tan- 
gent; the tangent where y" = is called a stationary tangent. 

Note 2. If f(x) is a rational integral function of degree n, the greatest 
number of points of inflexion that the curve y = f(x) can have is n — 2. 
Moreover the points of inflexion occur between points of maxima and minima. 
[See F. G. Taylor's Calculus (Longmans, Green & Co.), Art. 206.] 

Xote 3. References for collateral reading. On maxima and minima of 
functions of one variable, etc. : McMahon and Snyder, Diff. Gal., Chap. VI. ; 
Echols, Calculus, Chap. VIII. (in particular, Art. 85). On points of inflexion : 
Williamson, Diff. Cal. (7th ed.), Arts. 221-224 ; Edwards, Treatise on Diff. 
Cal, Arts. 274-279 ; Echols, Calculus, Chap. XL 

Kote 4. Points of inflexion : polar coordinates. Eor a discussion of 
this topic see Todhunter, Dip. Cal., Art. 294; Williamson, Diff. Cal., 
Art. 242; F. G. Taylor, Calculus, Art. 276. 

EXAMPLES. 

1. In the following curves find the points of inflexion, and write the 
equations of the inflexional tangents ; also sketch the curves and draw the 
inflexional tangents : (1) y = x s ; (2) x — 3 = {y + 3) 3 ; (3) y = x 2 (4 — x) ; 

(4) 12y = x*-6x 2 + 48; (5) ?/=_§_; (6) y =-A*-; (7) y- 



x 2 + 4 1 + x 2 4 + x 2 

2. Find the points of inflexion on the following curves : (1) y = 
x(x - ay ; (2) xy 2 = a 2 (a - x) ; (3) ax 2 - x 2 y -a 2 y = 0; (4) y = b + 

(c-z) 3 ; (5) y = m- b(x - c)% ; (6) £ 3 - 3 bx 2 + a 2 y = 0. 

3. Show that the curve y = x i has no point of inflexion. Sketch the 
curve. 

4. Show that the points where the curve y — b sin - meets the #-axis 
are all points of inflexion. a 

5. Show that the curve (1 + x 2 )y = 1 —x has three points of inflexion, 
and that they lie in a straight line. 

6. Show why a conic section cannot have a point of inflexion. 

7. Show, both geometrically and analytically, why points of inflexion 
may be called points of maximum or minimum slope. 



CHAPTER VIII. 

DIFFERENTIATION OF FUNCTIONS OF SEVERAL 
VARIABLES. 



H. B. This chapter may be studied immediately after Chapter VII., or its 
study may be postponed and taken up after any one of Chapters IX.-XVTL* 

79. Partial derivatives. Notation. Thus far functions of one 
independent variable have been treated; functions of two and 
of more than two independent variables will now be considered. 

Let u=f(x, y) (1) 

^n which f(x, y) is a continuous function (see Note 2) of two 
independent variables x and y. The value of the function for a 
pair of values of x and y is obtained by substituting these values 
in f(x, y). 

Thus, if f(x, y) = 3 x - 2 y + 7, /(l, 2) = 3 • 1 -^ 2 • 2 + 7 == 6. 

Note 1. Geometrical 
representation of a func- 
tion of two variables. 

The student knows how a 
continuous function of one 
variable can be represented 
by a curve. A continuous 
function of two variables 
can be represented by a sur- 
face. Thus the function z, 

when z =f( X ,y), (2) 

is represented by the sur- 
face LEGS if MP, the per- 
pendicular to the xy-plane 
erected at any point M (x, y) 
on that plane and drawn to 
meet the surface at P, is 
equal to/(x, y). 




Fig. 33. 



* See the order of the topics in Echols' Calculus. 
128 



79.] PABTIAL DERIVATIVES. 129 

References for collateral reading. See chapters on the geometiy of 
three dimensions in text-books on Analytic Geometry, for instance, those of 
Tanner and Allen, Ashton, Wentworth ; also Echols' Calculus, Chap. XXIV. 

Note 2. Continuous function of two variables defined. A function 
f(x, y) is said to he a continuous function of x and y within a certain range 
of values of x and y, when : (i) /(x, y) does not become infinitely great, and 
(ii) if, (a, b) and (a + h, b + k) being any values of (x, y) within this 
range, f{a + h, b + k) can be made to approach as nearly as one pleases to 
/(«, b) by diminishing h and k, and if /(a + h, b + k) becomes equal to f(a, 6), 
no matter in what way h and k approach to, and become equal to, zero. 
This definition may be illustrated geometrically, thus : On the x?/-plane 
(Fig. 83) let M be (a, 6) and N be (a + h, b + k), and let MP be /(a, 6) 
and XQ be /(a + A, 6 + jfe). Then, if jtfP and A r # are finite, and if XQ 
remains finite while A 7 " approaches M, and becomes equal to MP when X 
reaches M, no matter by what path of approach on the ary-plane, /(#, y) is 
said to be a continuous function of x and y f or x = a and y = b. 

In (1) suppose that # receives a change Ax and that y remains 
unchanged. Then u receives a corresponding change Aw, and 

u + Aw = f(x + A*a, ?/) ; 

and Am = /(a; + Aa, v) - /(«, y). 

._ Am __ /(a? + As, y)-f(x, y) } 
Ax Aa; 

and Iim^ ** = lim A ^ /(» + A«, y) -/(a, y) . 

Aa; Aa; 

This limiting value is called the partial derivative of u with 
respect to x, because there is a like derivative of u with respect 

tow,namelv, lim^ ** = lim^ & y ± A ^ =/fa y > - 
A?/ Am 

These partial derivatives are usually written 

a«, 3m, (3) 



respectively, in order to distinguish them from derivatives (like 

— , — , — , and so on) of functions of a single variable and from 
dx dy dt 

what are called total derivatives (see Art. 81). If u =f(x, y, z), 



130 DIFFERENTIAL CALCULUS. [Ch. VIII. 

the partial derivatives of the first order are — , — , and « 

dx dy dz 

According to the above definition, the partial derivative with 
respect to each variable is obtained by differentiating the func- 
tion as if the other variable were constant. Notation (3) is very 
commonly used, but various other symbols for partial derivatives 
are also employed. 

Note 3. Geometrical representation of partial derivatives of a func- 
tion of two variables. Let /(x, y) be represented by the surface LEGS 
(Fig. 33) whose equation is _ - , >. 

z — j \ x i y)' 

Take P any point (x, y, z) on this surface. Through P pass planes parallel 
to the planes ZOXand ZOY, and let them intersect the surface in the curves 
LFG and EPS respectively. Along EPS, x remains constant; and along 
LPG, y remains constant. Accordingly, from the definition above and 

r)z 

Art. 24 the partial x-derivative — is the slope of LPG at P, and the 

dz dx 

partial ^/-derivative 2- is the slope of EPS at P. 

dy 

EXAMPLES. 

1. If u = x s + 2 x-y + xy z + y 4 + e x + x cos y, 

rill 

then ¥— = 3 x 2 + 4 xy + y s + e x + cos y, 

dx 

and ^ = 2 x 2 + 3 xy 2 + 4 w» - x sin y. 

dy 

2. Find ^, ^, and 2M, when w=a; 8 +2tf 2 +3s 2 +e !B sin w+coszcosy. 

dx dy dz 

3. On the ellipsoid — + ^ + - = 1 : (a) find ^ and ^ at the point 

F 16 T 25 T 9 V ; 5x dy 

where x = 1 and y = 4 ; (&) find — and — at the point where y = 2 and 

a a dz dy 

z = 2 ; (c) find ^ and ^ at the point where z = 1 and x = 3. Make 

dz dx 

figures for (a), (&), and (c), and show what these partial derivatives repre- 
sent on the ellipsoid. 

4. Verify the following : 

(i) If u = \og(e* + ev), |if + |H = i ; 
dx dy 

(ii) if u = -^-, ^ + ^ = (x + 2,-l)«; 

(iii) If m = x^, x^ + y^ = (x + y + log u)u. 
dx dy 



80.] SUCCESSIVE PARTIAL DERIVATIVES. 131 

80. Successive partial derivatives. The partial derivatives of 
the first order described in Art. 79 are, in general, also continuous 
functions of the variables, and their partial derivatives may also 
be required. In the successive differentiation of functions of two 
or more variables, the following is one of the systems of notation : 



d /diA • .,, d 2 u 
— f — is written — ; 
dx \dxj dx 2 


dy 


fdu\ 


-,, d 2 u 
is written — ; 

dy 2 ' 


d fdu\ • .ji d 2 u 

— ( — is written ; 

dy \dxj dy dx 


dx 


fdu\ 

\<>y) 


f\2 

is written — — ; 
dxdy 


d f dhi V -4-4- &u 
— ( ■ ■ is written ■ : 

dz\dydxy dzdydx 


h 


fSSCs 


d u 

is written ; 

dzdx 2 ' 


d f d 2 u \ • ... d s u 

— is written : 

dz \dx dz) dz dx dz 


-i 


'd 2 u\ 


... dhi 

is written ; 

dzdy 2 ' 


and so on. 









Note 1. In this notation the symbol above the horizontal bar indicates 
the order of the derivative, and the symbols below the bar, taken from right 
to left, indicate the order in which the successive differentiations are to be 

performed. Thus — ^-^ — means that u is to be differentiated three times 
dx 2 dy dz s 

in succession with respect to z, and the result is then to be differentiated 

with respect to y ; and the function thus obtained is then to be differentiated 

twice in succession with respect to x. 

Note 2. The adoption, by mathematicians, of the symbol d in the nota- 
tion of partial differentiation was mainly due to the great mathematician, 
Carl Gustav Jacob Jacobi (1804-1851), who decided, in 1841, to use d in 
the manner which afterwards became the fashion. As to some points of 
insufficiency and difficulty connected with this notation, see correspondence 
between Thomas Muir and John Perry, Nature, Vol. 66, pages 53, 271, 520. 

Note 3. The order in which the successive differentiations are per- 
formed does not affect the result (certain conditions being satisfied); e.g. 

d 2 u _ d 2 u d 3 u __ d 3 n __ g% d*u = d B u = d*u _ 

dxdy dydx dxdydz dzdxdy dydxdz dzdxdz dz 2 dx dxdz 2 

This theorem is true in almost all cases which occur in practice ; e. g. see Exs. 1-8. 
For a discussion and references see Infin. Cal., Art. 85. Also see Pierpont, 
Functions of Real Variables, Vol. L, Art. 418, and Gibson, Calculus, page 221. 



132 DIFFERENTIAL CALCULUS. [Ch. VIII. 



EXAMPLES. 

1. Show that — 2 — (^4x m ?/ n ) = — - — (Ax m y n ), in which .4, w, and n 

dx dy dy dx fl2 fl2 

are constants. Then show that if u = I,Ax m y n , — — = — , and hence 

dx dy dy dx 
that the theorem in Note 3 is true for all algebraic functions. 

f) 2 U d 2 U 

2. In the following instances verify the fact that — — - = — — ; 

v av-bx dxdy dydx 

u = sin (xy) ; u = cos £ t u = xM ; w = -2 ; m = sec {ax + by) ; u = xlogy; 

x by - ax 

u = x sin y -f ?/ sin x ; m = y log (1 + x#) ; w = sin (ay) ; u — sin (x)». 

3. In the following instances verify that dHl ■ 



f „\ r .,, d# 2 d& dydxdy dxdy 2 

(i) m = a tan- 1 M £ J ; (ii) u = sin (xy) + — 



x# 

. when 
dx 2 dy 2 dy 2 dx 2 



4. Show that dHl = d w , when m = cos (ax» + &y m ). 



5. If it = tan (y + ax) + (« - ax) *, show that ^ = a 2 ^- 

d# 2 6V 

6. If M = ^L, show that x2» + y_2!L=a5« J and that y^ + 

x + 2/ 5x 2 ^3x3^ Qx dy 2 

d' 2 u _ o d?*. 
dx dy dy 



3 2 M, 9wi d 2 « J _„ 2 3%__2 



7. If m = Vx 2 + y 2 , show that x 2 2-2 + 2 x?/ -2-!L + ^2 aj? = _ t w . 

dx 2 dx6ty d*/ 2 9 

8. Iiu= (x 2 + y 2 + 2 2 )-*, show that & + & + & = <>. 

dx 2 dy 2 dz 2 

9. Show that a function of two independent variables has n + 1 partial 
derivatives of order n. 

8L Total rate of variation of a function of two or more variables. 

N.B. Before reading this article and the next it is advisable to review 
Arts. 25, 26. 

Given that u =f(x, y), (1) 

and that x and y vary independently of each other, it is required 
to find the rate of variation of u in terms of the rates of variation 

of x and y ; i.e. to find — in terms of — and -^« 
dt dt dt 

In (1) let x and y receive increments Ax and Ay respectively, in 
a time At say ; then u receives a corresponding increment Au, and 

u + Au =f(x + A.t, y + A?/). 

.-. Am =/(a> + Aa>, y + Ay) -/(aj, y). (2) 



81.] TOTAL BATE OF VARIATION. 133 

Hence, on introduction of — /(ft, y -f- Ay) +/(ft, y -}- Ay) and 
division by At, 

Au = f(x + Ax, y + Ay) -/(a?, y + Ay) f(x, y + Ay) -/(ft, y) . 
Af At At 

_ f(x+Ax,y+Ay)-f(x,y+Ay) Ax f(x,y+Ay)-f(x,y) Ay 
Ax At Ay At 

Now let At approach, zero ; then Ax and Ay approach zero, and, 
moreover (if a certain condition is satisfied), 

li™ f(x + Aa;, y + Ay) -/(a, y + Ay) _ df(x, y) * . du . 



and lim 



Ay===0 



/(ft, y-\-Ay)-f(x, y) = 3m 
Ay dy 



Hence, *! = ** ^ + ^ M. (3) 

• ' dt doc dt dy dt w 

In words : 77ie total rate of variation of a function of x and y is 
equal to the partial x-derivative multiplied by the rate of variation of 
ft plus the partial y-derivative multiplied by the rate of variation of y. 

Similarly, if u =f(x, y, z), 

du _ du dx du dy du dz ... 

dt dx dt dy dt dz dt 

Results (3) and (4) can be extended to functions of any num- 
ber of variables. (All derivatives herein are assumed to be con- 
tinuous.) 

Note 1. A function may remain constant while its variables change. 
The total rate of variation of such a function is evidently zero. (See Art. 84.) 

Note 2. Suppose that in (1) y is a function of x and that the derivative 
of u with respect to x is required. This may be obtained either directly, as 
(3) has been obtained, or by substituting x for t in (3) ; then 

du = du,dud]l t /g\ 

doc doc dy doc 

dx 
Result (5) may also be obtained by dividing both members of (3) by — 

[Art. 34(3)]. dt 

* For a discussion of the condition necessary and sufficient for the passage 
of the first member of this equation into the second, see W. B. Smith, Infini- 
tesimal Analysis, Vol. I, Art. 205 (and also Arts. 206, 207). 



134 DIFFERENTIAL CALCULUS. [Ch. VIII. 

du 
Note 3. In (5) ^— is the ^-derivative of u when y is treated as a con- 
stant, and — is the ^-derivative of u when y is treated as a function of se. 
dx 

Here — is called the total ^-derivative of u, 

dx 

Similarly the total w-derivative §Ji = du + du^. 

dy dy dx dy 



EXAMPLES. 

1. Express result (5) in words. 

2. Given z = 3x' 2 + 4y 2 , (1) 

find — whence =3, ?/=— 4, — = 2 units per second, and -^ = 3 units per second. 
dt dt dt 

On differentiation in (1), ^ = 6x — + 8y- y - = -60. 
K ' dt dt dt 

Geometrically this means that on the surface (1), which is an elliptic 
paraboloid, if a point moves through the point (3, — 4, 91) in such a way 
that the x and y coordinates of the moving point are there increasing at the 
rates of 2 and 3 units per second respectively, then the ^-coordinate of the 
moving point is, at the same place and moment, decreasing at the rate of 
60 units per second. 

N.B. Figures should be drawn for Ex. 2 and the following examples. 

3. In Ex. 3(a), Art. 79, find how the ^-coordinate is changing when 
the ^-coordinate is increasing at the rate of 1 unit per second, and the 
y-coordinate is decreasing at the rate of 2 units per second. 

4. In Ex. 3 (&), Art. 79, find how x is behaving when y is decreasing 
at the rate of 2 units per second, and z is increasing at the rate of 3 units 
per second. 

82. Total differential. Let dx and dy be differentials of the x 
and y in (1) Art. 81. They may be regarded as quantities such that 

dx:dy = ^-M- 

dt dt n 

Now let du be taken so that 

du = ^dx + ^dy. (1) 

doc dy v J 

As used in (1) -^ dx is called the partial x-differential ofu, — dy 
ox dy 

is called the partial y-differential of u, and du is called the total 
differential of u, and the complete differential of u. 



82.] TOTAL BIFFEBENTIAL. 135 

Note 1. When y is a function of x, relation (1) follows directly from 
Eq. (5), Art. 81, and definition (5), Art. 27. 

Note 2. The partial differentials in (1) are also denoted by d x it and d y u, 
and thus (1) may be written ^ = ^ + ^ 

Note 3. In general the du in (1) is not exactly equal to the actual change 
in u due to the changes dx and dy in x and y ; but the smaller dx and dy are 
taken, the more nearly is du equal to the real change in u (see exercises below). 
The differential du may be regarded as, and is very useful as, an approxima- 
tion to the actual change in u. In some cases this change can be calculated 
directly ; in others it can be found to. as close an approximation as one pleases 
by a series developed by means of the calculus. [See Chap. XVI., in par- 
ticular, Art. 150, Eq. (10), and Art. 152, Note 5.] 

EXAMPLES. 

1. Express relation (1) in words. 

2. Given u = 3 x 2 + 2 y 2 , find du when x = 2, y = 3, dx = .01, and 
dy = .02. 

Here du = 6 x dx + 4 ydy = .12 + .24 = .36. 

The actual change in u is 3(2 . 01) 2 + 2(3 • 02) 2 - (3 • 2 2 + 2 . 3 2 ) = .3611. 

3. As in Ex. 2 when dx = .001 and dy = .002. Also find the change in u. 

4. Eind the complete differential of each of the following functions : 

(i) tan -1 ^; (ii) y x ; (iii) xv \ (iv) loga^; (v) u = x l °sv. 

5. Eind dy when y = 8 cos A sin B, A = 40°, dA = 30', B = 65°, 
dB = 20'. 

Note 4. It may be said here that if LB OS (Fig. 38) be the surface 
z = /(x, y) , and if M be (jc, ?/) and A 7 " be (x + $£, ?/ + dy) , and A 7 ^ be pro- 
duced to meet in $i the plane tangent to the surface at P, then the total 
differential dz is equal to JSfQi — MB. 

Ex. Prove this statement. (Suggestion : make a good figure.) 

Similarly to (1), if u =f(x, y, z), and dx, dy, dz, be differentials 
of x, y, z, respectively, and if du be taken so that 

du = ^dx + ^dy + ^d» 9 (2) 

doc dy dz ^ } 

du is called the total differential of u. Eelation (2) is also written 

du = d x u + d y u + d z u. 

Definitions (1) and (2) may be extended to functions of any 
number of variables. 



136 DIFFERENTIAL CALCULUS. [Ch. VIII. 

6. Given u = x 2 + y 2 + 2 s, find du when x = 2, y = 3, = 4, dx = .1, 
dy = .4, cte = — .3. Also find the actual change in u. 

7. The numbers u, x, y, and z being as in Ex. 6, dx = .01, cfy = .04, and 
dz = — .03, calculate the difference between du and the actual change in u. 

8. Find du when w = xv z . 

83. Approximate value of small errors. A practical application 
of relations (1) and (2), Art. 82, may be made to the calculation 
of approximate values of small errors. The ideas set forth in the 
first part of Art. 65 may be applied to any number of variables. 

If u = f(x,y,z,—), 

and dx, dy, dz, '••, be regarded as errors in the assigned or measured 
values of x, y, z, •••, then 

, du -, , du , . du -, . 

du = — dx H dy -\ dz+ ~- 

ox oy dz 

is, approximately, the value of the consequent error in the com- 
puted value of u. Illustrations can be obtained by adapting 
Exs. 2, 3, 5, 6, 7, Art. 82. In applying the calculus to the com- 
putation of approximate values of errors it is usual to denote the 
errors (or differences) in u,x,y, •••, by Au, Ax, A?/, •• rather than 
by du, dx, dy, •••. Other notations are also used ; e.g. hu, Sx, &y, •••. 

EXAMPLES. 

1. In the cylinder in Ex. 3, Art. 65, give an approximate value of the 
error in the computed volume due to errors Aft in the height and Ar in 
the radius. 

Let V denote the volume. Then V = wr 2 h. 

:. AV = 2 rrrh • Ar + nr 2 • Ah. 

The relative error is — = ^T + ^ . 
V r h 

2. Do as in Ex. 1 for a few concrete cases, and compare the above 
approximate value of the error with the actual error. What is the difference 
between the actual error in the volume in Ex. 1 and its approximate value 
obtained by the method above ? 

3. In the triangle in Ex. 7, Art. 66, let Aa, Ab, AC, be small errors 
made in the measurement of a, b, C : show that the approximate relative 

error for the computed area A is — + — + cot C • A C. 

a b 



83,84.] IMPLICIT FUNCTIONS. 137 

Find, by the calculus, an approximate value of AA, given that a = 20 inches, 
b = 35 inches, C = 48° 30', A« = .2 inch, Ab = .1 inch, AC = 20'. How can 
the actual error in the computed area be obtained ? 

4. Show that for the area A of an ellipse when small errors are made 

in the semiaxes a and 5, approximately _ = ^ + ^. 

A a b 

In this general case, and in several concrete cases, compare the approxi- 
mate error in the computed area with the actual error. 

5. In the case described in Ex. 3 show that if Ac denote the consequent 
error in the computed value of c, then, approximately, 

Ac = cos B • Aa + cos A • Ab + a sin B - AC. 

N.B. For remarks and examples on this topic see Lamb, Calculus, 
pp. 138-142, Gibson, Calculus, pp. 258-260. 

84. Differentiation of implicit functions, two variables. This 
topic has been taken up in one way in Art. 56. Let the relation 
connecting two variables x and y be in the implicit form 

M y) = c, (i) 

in which c denotes any constant, including zero. Let u denote the 
function f(x, y) ; then (1) may be written 

u = c. (2) 

Since u remains constant when x and y change, — = ; i.e. 
(Art. 81, Eq. 3, and Note 1) dt 

dn dx dudy_ r, /o\ 

dx~dt dy dt~ ' 

dy du $u 

From (3), | = - g; whence [Art. 34, Eq. (3)], % = - g. (4) 

dt By dy 

Ex. 1. Express relation (4) in Avords. 

Xote. It should not be forgotten that the relation between the function 
and the variable should be expressed in form (1) before (4) is applied. 

Ex. 2. Do Exs. 13, 14, Art. 37, and exercises, Art. 56, by the method of 
this article, Compare the methods of Arts. 37, 56, and 84. 



138 DIFFERENTIAL CALCULUS. [Ch. VIII. 

85. Condition that an expression of the form Pdx + Qdy be a total 
differential. This article may be regarded as supplementary to 
Art. 82. 

Suppose that f x (x, y) and f 2 (x, y) are two arbitrarily chosen 
functions : does a function exist which has f x (x, y) for its partial 
cc-derivative and f 2 (x, y) for its partial ^-derivative ? A little 
thought leads to the conclusion that in general such a function does 
not exist. The condition that must be satisfied in order that there 
may be such a function will now be found. Suppose that there is 
such a function, and let it be denoted by u. Then, according to 
the hypothesis, 

-^=/i(>, V) and y=f*(x, y). (1) 

By Art. 80, Note 3, -^- = -^- ■ (2) 

J ; ' dydx dxdy w 

Hence, from (1) and (2), 

Eesult (3) is directly applicable to the differential expression 
Pdx + Qdy on substituting P for f^x, y) and Q for f 2 (x, y). 
Otherwise : If Pdx 4- Qdy is a total differential, du say, then 

^ = Pand^ = Q. (4) 

dx oy 

Hence, from (2) and (4), !** = $£.,. (5) 

dy doc 

When condition (5) is satisfied, Pdx + Qdy is also called an 
exact differential. 

Note 1. That this condition is not only necessary (as shown above), but 
also sufficient, is shown in works on Differential Equations. {E.g. see 
Professor McMahon's proof in Murray, Diff. Eqs., Note E.) 

Note 2. Eor the condition that an expression of the form Pdx + Qdy 
4- Bdz (see Art. 82, Eq. 2) be a total differential, see works on Differential 
Equations ; e.g. Murray, Diff. Eqs., Art. 102 and Art. 103, Note. 



85, 86.] PARTIAL DIFFERENTIALS. 139 

Ex. 1. Apply test (5) in the following cases : (a) u = 3 x 2 + 2 y 2 ; 

(&) w = tan^; (c) xdy -{-ydx; (d) xdy — ydx. 
x 

Ex. 2. Illustrate by examples the phrase, " in general such a function 
does not exist," which occurs in this article. 

Note 3. On Eider'' s theorem on homogeneous equations and successive 
total derivatives see Infin. Calculus, Arts. 87, 88. 

86. Illustrations: partial differentials, total differentials, partial 
derivatives. Illustrations of partial derivatives have already been 
given in Art. 79, Note 3. Partial differentiation is often required 
in engineering, physics, and other sciences. Accordingly, a stu- 
dent should try to get a good understanding of the subject. The 

interesting and peculiar relation h ®__ F 

shown in Illustration impresses § 
the necessity of having clearly in 
mind the conditions under which ^ 
a partial derivative is obtained. 

Illustration A. Suppose that ,_y_ 

OABC is a rectangular plate ex- "^U * -^dx*\~ 

panding under the application of Fj g- 34. 

heat. Let x, y, denote its sides and u its area. 

Then u = xy. (1) 

From (1), on taking the partial derivatives (Art. 79), 

dx ' dy 

.-.du = ~dx + — dy [Eq. (1), Art. 821 
dx dy 

= ydx + xdy. (3) 

In Fig. 34, AD, CH, denote dx, dy, the differentials of the 
sides x, y ; 

the partial a>diff erential of the area is ydx, i.e. BD ; 
the partial ?/-diff erential of the area is xdy, i.e. HB ; 
the total differential of the area = ydx + xdy = BD + HB. 
The difference in area = BD + HB + GE. 
See Art. 82, Exs. 2-5. 



140 



DIFFEBEN TIAL CALCUL US. 



[Ch. VIII. 




87. Illustration B. 

Note. In the case of a function y =/(x), 

Draw the curve y =f(x), and at any point 
P(x, y) draw the tangent FT. 
Draw FS parallel to OX. 
Then 

t&n SFB=- J -. 
dx 

Let NM = dx, and draw the ordinate MQ meeting the tangent at B. 
Then SB = FS tan SFB = f (x) • dx. 

Hence SB = dy, 

and thus, as pointed out in Art. 27, Note 1, dy is the increment in the ordi- 
nate drawn to the tangent corresponding to an increment dx in the abscissa. 

At any point P(x, y, z) on a 
surface 

z=f(x,y) (1) 

let the tangent plane PSQR 
be drawn. Draw PN parallel 
to OZ meeting the a?2/-plane 
in N(x, y). Now suppose 
that x, y receive increments 
dx and dy, as indicated in the 
figure NLMG. 

Draw LG, NM, meeting 
in V. Through L, 31, G, V, 
draw lines parallel to OZ and 
meeting the tangent plane in 
R, Q, S, C, respectively. 

Through P pass the plane PFKH parallel to XOY. 

By Art. 79, Note 3, tan FPR = — , tan HPS = — • 

dy dx 

Here NP = z; MQ = MK+ KQ = NP + KQ = z + KQ; 

GS = GH+ HS = NP+PHt3Ln HPS = z + — dx-, 

dx 

LR = LF+ FR = NP+PFtanFPR = z+ — dy. 

dy 




{x + dx, y + dy) 



88.] 



Now 



i.e. 



PARTIAL DIFFERENTIALS. 
GV= NP±MQ. also cr= GS + LB. 

dz dz 

z + z + KQ = z + — dx-\-z + — dy. 
ox dy 

.-. KQ = -^-dx + ^-dy. 
ax oy 



141 



But, from (1) by definition, Art. 82, 
dz 



dz , . dz , 

— dx-\ dy. 

dx dy 



.-. dz = KQ. 
That is, if the surface z—f{x, y) be described, and a tan- 
gent plane be drawn at a point (x. y, z), dz is the increment 
in the length of the ordinate drawn to the tangent plane from the 
a?/-plane when increments dx and dy are given to x and y. 

88. Illustration C. In Fig. 37 let P 
be the position of a moving point at any 
instant, and let its rectangular and polar 
coordinates, chosen in the ordinary way, 
be (x, y), (r, 0), respectively. The 
following relations hold : 

x = r cos 0, 

,2 1 „,2 



8xN 




Fig. 37. 



(i) 

(2) 



(either severally or all), 



r = xr -+- y 
When the point P moves, x, y, r, 
change. 

Note. Occasionally it is necessary to indicate the variable which is re- 
garded as constant when a partial derivative is obtained. For this the fol- 
lowing notation is sometimes employed : 

The partial derivative of x with respect to r, 6 being kept constant, is 



written 



dr 



the partial derivative of x with respect to r, 



written 



fjr 



.,;- ~ (<M\ =COS = ^. 



From (2), by Art. 79, 



being kept constant, is 

(3) 
(4) 



142 DIFFERENTIAL CALCULUS. [Ch. VIII. 

Hence, from (3) and (4), in the case of a point moving in a 

plane (S) e =7^r (5) 



dr. 

That is : the partial derivative of the abscissa with respect to the 
distance when the argument* is kept constant, 
is the reciprocal of 

the partial derivative of the abscissa with respect to the distance 
when the ordinate is kept constant. 

This is a curious instance in which the partial derivative of one 
variable with respect to a second under one condition, is the recip- 
rocal of the partial derivative of the same variable with respect 
to that second under another condition. 

Geometric treatment of Illustration C. Relations (3) and (4), 
from which (5) follows, can be shown geometrically. 

In Fig. 37 suppose that P moves to P ]? say, 6 being kept constant. 
Then r and x change by the amounts PP 1 and PN respectively. 

Then in PP.N, cos 6 = lim^/gj = gV (6) 

Now suppose that P moves to P 2 , say, y being kept constant. 
Through P 2 describe a circular arc about 0, cutting OP in M. 
Then r and x change by the amounts PM, PP.,, respectively. 
Then, in a manner similar to that taken in Art. 63, it can be 

shown that cos — \im PP J — — j = ( — ^ ) • (7) 



y 



Hence, from (6) and (7), (g^=g^= * 

[dr 
EXAMPLES. 

1. Given that (x, y), (r, 6) are the corresponding rectangular and polar 
coordinates of a point P, show : 

(a) (dx) 2 + {dy) 2 = (dr) 2 + r 2 (<Z0) 2 > 
(6) xdy — ydx = r 2 dd. 
[Suggestion, x = r cos 0, y = r sin 6 ; see Art. 82, Eq. (1) ]. 

2. Construct figures representing relations (a), (6), in Ex. 1. 

* ' The angle ' in the case of a point P (r, 0) is called ; the argument 
of P.' 



CHAPTER IX. 

CHANGE OF VARIABLE. 

N.B. If it is thought desirable, the study of this chapter may be post- 
poned until some of the following chapters are read. 

89. Change of variable. It is sometimes advisable to change 
either, or both, of the variables in a derivative. If the relation 
between the old and the new variables is known, the given 
derivative can be expressed in terms of derivatives involving the 
new variable, or variables. Arts. 91-93 are concerned with 
showing how this may be done. In Art. 90 an expression for the 
given derivative is found when the dependent and independent 
variables are interchanged ; in Art. 91, when the dependent 
variable is changed; in Art. 92, when the independent variable 
is changed; and in Art. 93, when both the dependent and the 
independent variables are expressed in terms of a single new 
variable. In N ote 1, Art. 93, an example is worked in which the 
dependent and the independent variables are both expressed in 
terms of two new variables. 

N.B. Principle (2) of Art. 34 is repeatedly employed in Arts. 90-93. 

90. Interchange of the dependent and independent variables. Let 

y be the dependent and a; the independent variable. Also let y 
be a continuous, and either an increasing or a decreasing, function 
of x. 

Then Ay =£ when Ax =£ 0, and ^ = (1) 

* Aa? Aaj v J 

Ay 
Since y is continuous, Ay = when Aa; = ; accordingly from (1), 

dy 
143 



144 

Again, 



DIFFERENTIAL CALCULUS. 

<Py = d L fdy\ = _d fdy\ m dy / Art 34 x 
dx 2 dx\dx) dy\dxj dx 



[Ch. IX. 







d 2 x 


£.[i; 


dte 


dtf 


dy 1 dx 


' dy~ 


fdx\ 3 


l^J 





Ex. Express the third ^-derivative of y in terms of ^-derivatives of x. 

91. Change of the dependent variable. Let the dependent and 
independent variables be denoted by y and x respectively. It is 
required to express the successive derivatives of y with respect to 
x y in terms of the derivatives of z with respect to x when 

y = F(z). 



dy_dydz = F ,,, < dz _ 
dec dz dx dx 



_ d fdy\ _ d 



F'(z) — 1 
w dx] 



dx 2 dx\ dx) dx 

d 2 z , dz d TTTf/.x -r7,t/..\d 2 z , dz d r -wn/„\n dz 



= ^'(2)^ + ^1 . _^L F'(z) = F'(z) ^ + ,— • — [.F'OOl • 
^ J dx 2 dx dx W KJ dx 2 ^dx dz 1 WJ 

W dz 2 W UaJ 



da? 



Ex. 1. Given that y = F(z), show that 

i=^)i + 3^).g| + ^)(|y 

Ex. 2. Change the dependent variable from y to z in 

( l+ ^S- 2 *)+(I) 3 = 2 ^ 

given that y = z 2 + 2 2. 



dy d 2 y 
dx dx 2 ' 



From (2), 






:2(S + 1). 



Now d 1 = dydz [Art . 34(1)] =2(0 + 1)-- 

dx dz dx dx 

AH> ^-^( d y) = ±[2(z + 1)^1 = 2(0 + l)^ + 2 f^V 

die 2 dx\dx) dx L da; J dx 2 V^/ 

dx 3 dxXdx 1 / dx\_ dx 2 \dx) J 



cfa; 3 dx dx 2 



(1) 
(2) 
(3) 
(4) 

(5) 
(6) 



91, 92.] 



CHANGE OF VARIABLE. 



145 



Substitution in (1) of the values of y and its derivatives, from (2), (4), 
(5), (6), and reduction give 

v J dx* dxdx 2 



92. Change of the independent variable. Let the dependent and 
independent variables be denoted by y and x respectively. It is 
required to express the successive derivatives of y with respect to 
x, in terms of the derivatives of y with respect to z when 

x=f(z). 

1 



Here 



— = f'(z), and hence, — = — 
dz J w ' ' dx f(z) 

. dy _ dy dz _ 1 dy 
dx dz dx f'(z) dz 

d 2 y _ d fdy\ __ d (dy\ dz _ d f 1 dy 
dx 2 dx\dxj dz\dxj dx dz\f'(z)dz 



dz 

dx 



1 






' 1 d 2 y f"(z) dy 
f\z) ' dz 2 lf{z)-f ' dz_ 



Ex. 1. Find ^ when x = /(s). 
dx* 



Ex. 2. Change the independent variable from x to t in 

0, 



d 2 y 2x dy 



given that 
From (2), 



dx 2 l + x 2 dx (1 + z 2 ) 2 

x = tan t. 

— =sec 2 t; whence — = — — 
dt dx sec 2 t 

dy = dy.dt [Art 34 (1)] = _1_^ 
dx dt dx sec 2 tdt 



d 2 y _ d_ (dy 
dx 2 dx \dx 



d_fdy 
dt\dx 



dt^_ d_ I 1 dy 
dx dt\sec 2 tdt 



dt 
dx 



1 cPy _ 2tan£ dy\ 1 
sec 2 1 dt 2 sec 2 t dt J sec 2 



(1) 

(2) 
(3) 

(4) 



(5) 



Substitution in (1) of the values of x, ^, ^ from (2), (4), (5), and 

dx dx 2 



reduction give 



dt 2 



146 DIFFERENTIAL CALCULUS. [Ch. IX. 

93. Dependent and independent variables both expressed in terms 
of a single variable. 

Let y = ( ji(t) and x =f(t). 

Then dy = dy_^_dx .-^ ^ , g ^ = £(£) ^ 

dx dt dt l ' K JJ fit) 

dx 2 ~~ dx [dxj dt [dx] ' dx ~ dt |_/'(0 J ' /'(*) 

= /(0»''(0-*W"(0 
[/'(OP 

Similarly for higher derivatives. 

See Art. 71, which is practically the same as this, and its 
Exs. 1, 2. 

EXAMPLES. 

1 . In the above case find — ^ • 

2. Given that x = a{d — sin 0) and y = a(l — cos 0), calculate 

3. Given that £ = a cos 6 and ?/ = a sin d, calculate the same function as 
in Ex. 2. What curve is denoted by these equations ? 

4. Given that x = a cos 6 and y = b sin 0, calculate the same function as 
in Ex. 2. What curve is denoted by these equations ? 

Note 1. Both dependent and independent variables expressed in terms of 
two new variables. Following is an example of this. 

Ex. Given the transformation from rectangular to polar coordinates, viz. 

x = rcosd, y= r sin 0, (1) 

express -^ and — ^ in terms of r, 0, and the derivatives of r with respect to 0. 
dx dx 2 

From (1), ^ = cos0 — -rsin0, ^= sin — + rcos 0. 
dd dd dd dd 

dv 

sm0— + rcos0 
dy Idy dx . . \ dd 



-v- — , Art. 34, Eq 



•(3)) 



J ? cos0^-rsin0 

dd 



d (dy\ d (dy\ dd \dd ) dd 2 



\dx) 



dx 2 dx\dx dd\dxl dx I a dr 



cos d -jq — r sin 



<y 



93.] CHANGE OF VARIABLE. 147 

Note 2. For more complex cases of change of the variables in a deriva- 
tive, see other text-books. 

Note 3. References for collateral reading. Williamson, Diff. Cal., 
Chap. XXII. ; McMahon and Snyder, Diff. Cal., Chap. XI. ; Edwards, 
Treatise on Diff. Cal., Chap. XIX.; Gibson, Calculus, §§ 98, 99. 

EXAMPLES. 

X.B. In working these examples it is much better not to use the results 
or formulas derived in Arts. 90-93, but to employ the method by which these 
results have been obtained. - 

1 . Change the independent variable from x to y in : (i) — ^ + 2 x ( -^ ) = ; 

dx 2 \dxJ 

an 3ffty_^ft-ft W 2 =o 

k J \dx 2 ) dxdx* dx 2 \dx) 

2. In ft = 1 + 2 ( ] + y } l^-Y, change the dependent variable from y to 

dx 2 1+y 2 \dxj 

z, given that y — tan z. 

3. Change the independent variable under the following conditions : 

(i) x 2 p± + x ^ + u = 0, y = logx; (ii) (1 -a 2 )f^- x& + y = 0, x = cost; 
dx 2 dx dx 2 dx 

(iii) (1 - x 2 )ft - x ^- = 0,x = cost; (iv) x°-ft + 2 x^- + —y = 0, xz = 1 ; 

dx 2 cZx dx 2 dx x 2 

(v) x 3 ft + 3 x 2 ft + x^ + y = 0, s = log x ; (vi) x 4 ft + 6 x 3 ft + 9 x 2 ft 
w dx 3 dx 2 dx y s , k j dx ^ dx3 ^ dx2 

+ 3x^ + y = logx,x = e*. 
dx 

4. Find -^ and — ^ when : (i) x — a(cos t + £ sin £) , y = a(sin t — tcost); 

dx dx 2 

(ii) x = cot t, y = sin 3 1. 

5. If x ft - x -( %> Y+ & = 0, and x = ye*, show that y ft + ** = 0. 

dx 2 y\dx/ dx dy 2 dy 



CHAPTER X. 

CONCAVITY AND CONVEXITY. CONTACT AND CURVA- 
TURE. EVOLUTES AND INVOLUTES. 

94. Concavity and convexity of curves : rectangular coordinates. 

Definition. At a point on a cnrve the curve is said to be con- 
cave to a line (or to a point off the curve) when an infinitesimal arc 
containing the point lies between the tangent at the point and the 
given line (or point off the curve). If the tangent lies between 
the line (or point) and the infinitesimal arc, the arc there is said 
to be convex to the line (or point). 

Thus, in Fig. 50 a, at P the curve MN is concave to the line OX, and con- 
cave to the point A ; in Fig. 50 6, at Pi the curve MN is convex to the line 
OX, and convex to the point A. The arc on one side of a point of inflexion 
is concave to a given line (or point), and the arc on the other side of the 
point of inflexion is convex to this line (or point) (see Figs. 31 a, 6) . 

The curves passing through P and R have the concavity towards 
the a>axis, and the curves passing through Q and S are convex 

to the ic-axis. At P y is positive; 

and — ^ is negative, for -M. decreases 
dxr dx 

as a point moves along the curve 

towards the right through P. At R 

y is negative; and — ^ is positive, 
-j ax 

for -M. increases as a point moves 
dx 

along the curve towards the right through R. Hence, at points 
where a curve is concave to the x-axis y -=-^ is negative. A similar 
examination of the curves passing through Q and S shows that at 
points wliere a curve is convex to the x-axis y -p=| is positive. 

148 



94, 95.] 



CONTACT. 



149 



Ex. 1. Prove the theorem last stated. 



Ex. 2. Test or verify the above theorems and Note 1 in the case of a num- 
ber of the curves in the preceding chapters. 

Note 1. The curves passing through P and 8 are concave downwards, 

d~v 
and here — 2 is negative. The curves passing through B and Q are concave 

upwards, and here -^-| is positive. 

Note 2. A point where a curve stops bending in one direction and begins 
to bend in the opposite direction as at L, A, D, JET, G, P, Figs. 31 a, 6, 32, 
is called a point of inflexion. 



Note 3. A curve /(r, 6) = is concave or convex to the pole at the point 

, 0) according as u + — is positive or 
dd 2 

McMahon and Snyder, Biff. Cal. , Art. 144.) 



(r, 6) according as u + c -^ is positive or negative, u denoting -. (See 
dd 2 r 



95. Order of contact. If two curves, y = <f>(x) and y = f(x), 
intersect at a point at which x = a, as in Fig. 39 a, then cf>(a) =f(a) 
and <f>'(a) =£f'(a). If <f>(a) = f(a) and <£'(a) =/'(a), then the curves 
touch as in Fig. 39 b, and they are said to have contact of the first 
order, provided that <f>"(a) =£/"(a). If <f>(a) =/(«), <£'(a) =/'(«), 
and <£"(a) =/"(«), but <£'"(a) =fcf'"(a), then the curves are said to 



V=f(x) 




= 0(a>) 



have contact of the second order, as in Fig. 39 c. And, in general, 
if <fi(a) = /(«) and the respective successive derivatives of <f>(x) 
and /(a;) up to and including the nth, but not including the 
(n + l)th, are equal for x = a, then the curves are said to have con- 
tact of the nth order. Hence, in order to find the order of contact 
of two curves compare the respective successive derivatives of y 
for the two curves at the points through which both curves pass. 



150 DIFFERENTIAL CALCULUS. [Cn. X. 

Note 1. Another way of regarding contact is the following. In analytic 
geometry the tangent at P (Fig. 40 a) is defined as the limiting position 
which the secant PQ takes when PQ revolves about P until the point of 
intersection Q coincides with P. The line then has contact of the first order 
with the curve. This notion of points of intersection of a line and a curve 
becoming coincident will now be extended to curves in general. Two curves, 




Fig. 40 a. Fig. 40 6. Fig. 40 c. Fig. 40 d. 

C\ and C2 (Fig. 40 &), are said to intersect when they have a point, as P, in 
common. They are said to have contact of the first order at P when the 
curves (see Fig. 40 c) have been modified in such a way that a second point 

of intersection Q moves into coincidence with P. (The value of ~ at P is 

then the same for both curves, according to the definition of a tangent as 

given above.) The curves are said to have contact of the second order at P 

when the curves have been further modified in such a way that a third point 

of intersection It moves into coincidence with P and Q (see Fig. 40 d). (The 

d ( dv\ d 2 y 

value of — ^ , i.e. -y^, is then the same for both curves at P.) And, in 
dx\dxj dx 2 J 

general, the curves are said to have contact of the nth order at a point P when 

n + 1 of their points of intersection have moved into coincidence with P. 

(At P the respective derivatives of y up to the nth. are then the same for both 

curves.) See Echols, Calculus, Art. 98. 

Note 2. In general a straight line cannot have contact of an order higher 
than the first with a curve. For in order that a line have contact of the first 
order with a curve at a given point, the ordinates of the line and the curve 
must be equal there, and likewise their slopes ; thus two equations must be 
satisfied. These equations suffice to determine the two arbitrary constants 
appearing in the equation of a straight line. For example, if the line 
y — mx + b has contact of the first order with the curve y = f(x) at the point 
for which x — a, the following two equations are satisfied, viz. : 

f(a) = ma + b, f'(a) = m ; 

from these equations m and b can be found. 

This line and curve have contact of the second order in the particular (and 
exceptional) case in which f"(a) =0; consequently (Art. 78), if there is a 



95.] CONTACT. 151 

point of inflexion on the curve y =f(x) where x = a, the tangent there has 
contact of the second order. 

The theorem at the beginning of this note is also evident from geometrical 
considerations. Since, in general, a line can be passed through only two 
arbitrarily chosen points of a curve, it is to be expected from Note 1 that in 
general a line and a curve can have contact of the first order only. 

Note 3. In general, a circle cannot have contact of an order higher than 
the second with a curve. Tor in order that a circle have contact of the second 
order with a curve at a given point, three equations must be satisfied, and 
these equations just suffice to determine the three arbitrary constants that 
appear in the general equation of a circle [see Eq. (2), Art. 96]. This 
theorem is also evident from Note 1 and the fact that, in general, a circle can 
be passed through only three arbitrarily chosen points of a curve. (In a few 
very special instances a circle has contact of the third order with a curve. 
See Ex. 4, Art. 101 ) 

Note 4. It is shown in Art. 156 that when two curves have contact of an 
odd order, they do not cross *ach other at the point of contact ; but when they 
have contact of an even order, they do cross there. Illustrations : the tangent 
at an ordinary point on a curve, as shown in Figs. 15, 17 ; the tangent at a 
point of inflexion, as in Eigs. 26 a, 6, 31, 32 ; an ellipse and circles having 
contact of second order therewith (see Ex. 4, Art. 101). This theorem may 
also be deduced from geometry and the definitions given in Note 1. 

N.B. As far as possible make good figures showing the curves, lines, and 
points mentioned in the exercises in this chapter. 



EXAMPLES. 

1. Find the place and order of contact of (1) the curves y = x* and 

y = 6 ;c 2 - 9 x + 4 ; (2) the curves y = x 3 and y = 6 x 2 - 12 x + 8. 

2. Determine the parabola which has its axis parallel to the y-axis, passes 
through the point (0, 3), and has contact of the first order with the parab- 
ola y = 2 x 2 at the point (1, 2). 

3. What must be the value of a in order that the parabola y = x + 1 
+ «(x — l) 2 may have contact of the second order with the hyperbola 
xy = 3 x - 1 ? 

4. Find the parabola whose axis is parallel to the y-axis, and which has 
contact of the second order with the cubical parabola y = x 3 at the point 
(1, !)• 

5. Determine the parabola which has its axis parallel to the y-axis and has 
contact of the second order with the hyperbola xy = 1 at the point (1, 1). 



152 



DIFFER ENTIA L CALC UL US. 



[Ch. X. 



96. Osculating circle. It was pointed out in Art. 95, Note 3/ 

that contact of the second order is, in general, the closest contact 

that a circle can have with a 
curve. A circle having contact 
of the second order with a curve 
at a point is called the osculating 
circle at that point. 

In Fig. 41 PT is tangent to the 
curve C at P. Every circle which 
passes through P and has its cen- 
tre in the normal NM touches C 
at P. One of these circles has 
contact of the second order wiih. 
C at P; let this circle be denoted 

by K. All the other circles, infinite in number, in general have 

contact of the first order only. 

Osculating circle : rectangular coordinates. The radius and the 

centre of the osculating circle at any point P(x, y) on the curve 




Fig. 41. 



y=fX x ) 



(i) 



will now be obtained. Denote the centre and radius by (a, b) 
and r. Then the equation of the osculating circle at the point 
(x, y) is 



(X-a) 2 +(Y-b) 2 = r 2 . 



(2) 



For the moment, for the sake of distinction, x and y are used 
to denote the coordinates of a point on the curve, and X and Y 
are used to denote the coordinates of a point on the circle. Then 
at the point where the circle and the curve have contact of the 

second order 

. . , ... 

(3) 



dX dx 



d 2 Y 



X=x, Y=y, ^L±. = ^L, ^_>_ = <±jl, 

dX 2 



&y. 

dx 2 



From (2), on differentiating twice in succession, 

x - a+(r - 6) i= ' 



i+ 



(gy +(r _ 6) ^ =0 . 



(PY 
dX 2 



(4) 
(5) 



96, 97.] 



CONTACT. 



153 



and 



Y-b = - 



X-a 



MU 



dX 2 ' 



dry 

dX 



dY . d 1 T t 
dX ' dX 2 ' 



Accordingly, from (3), (2), (6), (7), 



Min 



and from (3), (6), (7), 






a-x — 



\dx) dy m 



1 + 






<Za? 



b = y + 



d*y 
dx* 



(6) 
(7) 

(8) 
(9) 



Note. For the osculating circle, polar coordinates being used, see Art. 
102, Note 2. 

Ex. 1. Determine the radius and the centre of the osculating circle for 
each of the curves in Ex. 1 (1), Art. 95, at their point of contact. 

Ex. 2. Do as in Ex. 1 for the curves Ex. 1 (2), Art. 95. 

97. The notion of curvature. Let the cnrves A, B, C, D have 
a common tangent PT at P. At the point P the curve A, to use 
the popular phrase, bends or curves more than the curves B, C, 
and D ; and D bends or curves less than the curves A, B, and C. 
These four curves evidently differ in the rate at 
which they bend, or turn away from the straight 
line PT, at P. These ideas are sometimes ex- 
pressed by saying that these curves differ in 
curvature at P, and that there A has the greatest 
and D the least curvature. In the case of two 
circles, say one with a radius of an inch and the 
other with a radius of a million miles, it is cus- 
tomary to say that the second circle has a small 
curvature, and that the first has a large curvature in comparison 
with the second. An inspection of a figure consisting of a circle 
and some of its tangents gives the impression that what is popu^ 
larly called the curvature is the same at all points of that circle. 




Fig. 42. 



154 



DIFFERENTIAL CALCULUS. 



[Ch. X. 



On the other hand, an inspection of an elongated ellipse gives 
the impression that the curvature is not the same at all points 
of that ellipse, although at two particular points, or at four 
particular points, it may be the same. Curvature will now be 
given a precise mathematical definition and its measurement 
will be explained. 

Ex. 1. Draw an ellipse, and find by inspection the points where the curva- 
ture is greatest and where it is least. Show how to obtain sets of four points 
on the ellipse Which have the same curvature. 

Ex. 2. Discuss a parabola and an hyperbola in the manner of Ex. 1. 

98. Total curvature. Average curvature. Curvature at a point. 

At A x the curve C has the direction A±T X , which makes the angle 

cf> 1 with the #-axis ; at A 2 the 
curve has the direction A 2 T 2 , 
which makes an angle <f> 2 with the 
aj-axis. The difference between 
these directions represents the 
angle by which the curve has 
changed its direction from the 
direction of the line A X T X in 
the interval of arc from A x to 
A 2 . This difference, namely, 
T X RT 2 or 4> 2 — <£ 1? is called the 
total curvature of the arc A X A 2 . 

The average curvature for this arc is 

($2 — <M ■*■ length of arc A X A 2 . 
(Here the angle is measured in radians.) 

Accordingly, if (Fig. 44) A<f> is the angle between the tangents 
at A and B, then A<£ is the total curva- Y 
ture of the arc AB ; if As is the length 

of the arc AB, then — * is the average 

As 

curvature of that arc. Now let B 
approach A. The arc As and the angle 
A<£ then become infinitesimal ; and, 




finally, when B reaches A, — * has the ^ 



As 




Fig. 44. 




98, 100.] CURVATURE. 155 

limitiDg value -2. The limit As ^ — at any point on a curve, i.e. 
ds As 

*Q there, is called the curvature of the curve at that point. (The 
ds 

phrase " curvature of a curve " means the curvature of the curve 
at a particular point.) In all curves, with the exception of 
straight lines and circles, the curvature, in general, varies from 
point to point. 

99. The curvature of a circle. Let A and B be two points on 
a circle having its centre at 0. In 
Fig. 45 the angle between the direc- 
tions of the tangents AT X and BT 2 is 
A<£, say. Let As denote the length of 
the arc AB. Then AOB= T 1 RT 2 =^c\>. 
Hence, by trigonometry, As = rA<£. 

From this, 

4i>__i. w h ence ^4-1. m 

As r ' wnence ds - r W FlG - 45 - 

That is, the curvature of a circle is constant and is the reciprocal 
of (the measure of) the radius. 

Note. When the radius increases beyond all bounds, the curvature 
approaches zero, and the circle approaches a straight line as its limiting 
position. When the radius decreases, the curvature increases ; as the radius 
approaches zero and the circle thus shrinks towards a point, the curvature 
approaches an infinitely great value. 

It is shown in Ex. 5, Art. 227, that all curves of constant curvature are 
circles. 

Ex. Compare the curvatures of circles of radii 2 inches, 2 feet, 5 yards, 
2 miles, 10 miles, 100 miles, and 1,000,000 miles. 

100. To find the curvature at any point of a curve: rectangular 

coordinates. Let the curve in Fig. 44 be y=f(x), and let its 
curvature at any point A(x, y) be required. Let k denote the 
curvature at A, and <jf> denote the angle which the tangent at A 
makes with the o>axis. Take an arc AB and denote its length 
by As, and denote the angle between the tangents at A and B by 
A<£. Then, by the definition in Art. 98, 

& = ^atA 
as 



156 DIFFERENTIAL CALCULUS. [Ch. X. 

Now (Art. 59), tan <f> = ~ .: <j> = tan" 1 ^. 



dx 2 
ds ~~ ds\ v%M ^ dx) ~ dx\ dx) ds ~ -. fdy\ 2 ' dx 



-, _d<j> __ d f _ x c?2/\ df _ x dy\ dx _ dx 2 _ ds 
A? — _ ■ — -=- tan — — ] — — — tan ~~ 



\dx 
% [Art, 67 c(2)], k = ^ (1) 



This, by (1) Art. 99 and (8) Art. 96, is the same as the curva- 
ture of the osculating circle. 

In order to find the curvature at a definite point (x iy y x ) it is 
only necessary to substitute the coordinates x 1} y v in the general 
result (1). 

Ex. 1. Compute and compare the curvatures of the two curves in Ex. 1 (1), 
Art. 95, at their point of contact. 

Ex. 2. Find the curvature of the curve y = x 3 — 2 x 2 + 7 x at the origin. 
Determine the radius and centre of its osculating circle ,at that point. 

101. The circle of curvature at any point on a curve : rectangular 
coordinates. The circle of curvature at a point on a curve is the 
circle which passes through the point and has the same tangent 
and the same curvature as the curve has there. The radius of 
this circle is called the radius of curvature at the point, and the 
centre of the circle is called the centre of curvature for the point. 

The radius of curvature. Let It denote the radius of curvature 
and (a, p) denote the centre of curvature for any point (x, y) on 
the curve y =/(«). Then it follows from Art. 99, and Art. 100, 
Eq. 1, that 3 

doc 2 

(That is, R is the value of this expression at that point.) 

Note 1. There is an infinite number of circles that can pass through a 
given point on a curve and have the same tangent as the curve has there but 
not the same curvature, and there is an infinite number of circles that can 



101.] 



CURVATURE. 



157 



pass through this point and have the same curvature but not the same tangent 
as the curve has there ; but there is only one circle passing through the point 
that has there both the same tangent and the same curvature as the curve. 

Ex. 1. Illustrate Note 1 by figures. 

The centre of curvature. Since at any point on a curve the circle 
of curvature and the curve have the same tangent and curvature, 

it follows that — and — *- are respectively the same for the circle 
dx dx 1 

and the curve at that point. Accordingly (Art. 95, Note 3) the 
circle of curvature has, in general,* contact of the second order 
with the curve, and thus (Art. 96) coincides with the osculating 
circle passing through the point. Accordingly (Art. 96, Eq. 9) 



1+ rm 

\doc 
d 2 y 
dx 2 



dy t 
'doc 9 



P = y + 



d 2 y 
doc 2 



(2) 



Note 2. The coordinates of the centre of curvature may also be obtained 
in the following manner. 

Let C be the centre of the circle of cur- 
vature of the curve PL at P, and let the 
tangent PT make the angle with the 
x-axis. Draw the ordinates PM and CiV, 
and draw PB parallel to OX. Let R 
denote the radius of curvature. Then 
dy 
dx 

In Fig. 88 



NCP = 0, and tan 



ON= OM- BP=x- Rsm<j> 



dx 




N/T M 
Fig. 46. 



HI)] 



i + 



4l& 

dx* 



Mm 



= X — 



Also, p = NC = MP + BC = y + R cos <j> = y +• 
The results for Fig. 88 are true for all figures. 



(dy\ 
\dxj 



dy 

dx) ^ dy^ 
d 2 y dx 

dx* 

2 



d 2 y 
dx* 



(3) 



(4) 



* For an exception see the circles of curvature at the ends of the axes of 
an ellipse. (See Ex. 4 following.) 



158 DIFFERENTIAL CALCULUS. [Ch. X. 

Ex. 2. Verify the last statement by drawing the radii of curvature at points 
on each side of points of maximum and minimum in the curves in Fig. 80 

and carefully noting the algebraic signs of -^ and ^ at these points. 

dx d 2 x 

Note 3. A glance at Fig. 38 shows that at P and B the normal (Art. 62) 
and the radius of curvature have the same direction, and at Q and S they 
have opposite directions. Hence (see Art. 94) the normal and the radius of 
curvature at a point on a curve have the same or opposite directions accord- 
ing as y — ^ there is respectively negative or positive. 
dx 2 

Note 4. At a point of inflexion, according to Art. 78, and Art. 100, Eq. (1), 
the curvature is zero. 

Note 5. A centre of curvature is the limiting position of the intersection 
of two infinitely near normals to the curve. For a consideration of this im- 
portant geometrical fact, see Williamson, Diff. Cal. (7th ed.), Art. 229; 
Lamb, Calculus, Art. 150 ; Gibson, Calculus, Art. 141. 



EXAMPLES. 

3. Find the radius of curvature and the centre of curvature at any point 
on the parabola y 2 = 4 px. What are they for the vertex ? 

Apply the general results just obtained to particular cases, by giving p par- 
ticular values, e.g. 1, 2, etc., and taking particular points on the curves, 
and make the corresponding figures. 

N.B. As in Ex. 3, apply the general results obtained in the following 
examples to particular cases. 

4. As in Ex. 3 for the ellipse b 2 x 2 + a 2 y 2 = a 2 b 2 . Find the radii of cur- 
vature at the ends of the axes. Show that this radius at an extremity of 
the major axis is equal to half the latus rectum. Illustrate Note 4, Art. 95, 
by drawing an ellipse and the circles of curvature at various points on it. 
Show that the circles of curvature for an ellipse, at the ends of the axes, have 
contact of the third order with the ellipse. 

5. Find the radius and centre of curvature at any point of each of the fol- 
lowing curves : (1) The hyperbola b 2 x 2 - a 2 y 2 = a 2 b 2 . (2) The hyperbola 

xy = a 2 . (3) The catenary y = - (e a + e a ). (4) The astroid x J + y* = a ¥ . 

A 
(5) The astroid x — a cos 3 0, y = a sin 3 0. (6) The semi-cubical parabola 
x s = ay 2 . (7) The curve x 2 y = a 2 (x — y) where x — a. (8) The cycloid 
x = a(d — sin 0), y = a(l — cos 0). In this cycloid show that the length of 
the radius of curvature at any point is twice the length of the normal. 

6. Find the radius of curvature at any point of each of the following 
curves : (1) The parabola Vx + Vy = Va. In this curve show that a + 13 — 
3(* + V)- (2) The cubical parabola a 2 y = x\ (3) The catenary of uniform 



102.] CURVATURE. 159 

strength y = c log sec ( - ). (4) The witch xy 2 = a 2 (a — x) at the vertex. 

(5) The parabola x = acot 2 ^, y = 2 acot^. (6) The ellipse # = acos0, 
y = b sin 0. (7) The hyperbola x = a sec 0, y = & tan 0. (8) The catenary 
x = a log (sec + tan 8) , y = a sec 0. 

102. The radius of curvature : polar coordinates. This can be deduced 
(a) directly from the definition of curvature (Art. 98) and the definition of 
radius of curvature (Art. 101) ; and (6) from form (1), Art. 101, by the 
usual substitution for transformation of coordinates, namely, x = r cos 0, 
y = r sin 9. 

(a) By Art. 63 (2), = + ^. 

Now k = & (Art. 98) = * • * = ( 1 + m l> + (^Vl~*. 

ds v y dd ds \ ddjl \ddj J 

[Art. 67 d % Eq. (3).] 



d9 
Also, tarn/' = r — (Art. 63). .*. -ty = tan - 



i MSW 



# _ \dej____de 2 
de- r2 ,d_r 
\de 



(6) The deduction of (1) from (1), Art. 101, by the transformation of coor- 
dinates is left as an exercise for the student. /7 \2-il 

Note 1. On the substitution of u for - in (1), R = - — - — J- 

r v J of . d 2 u 

V dd 2 

Note 2. Since the osculating circle and the circle of curvature coincide, 

the forms just found for R give the radius of the osculating circle. 

Note 3. For other expressions for R see Todhunter, Biff. Cal., Art. 321, 

and Ex. 4, page 352 ; Williamson, Biff. Cal. (7th ed.), Art. 236. Also see 

F. G. -Taylor, Calculus, Arts. 288-290. 

EXAMPLES. 

1. Find the radius of curvature at any point of each of the following 
curves : (1) The circles r = a and r = 2 b cos 0. (2) The parabola r(l + cos 0) 
= 2 a. (3) The cardioid r = a(l -f cos 0). (4) The equilateral hyperbola 
r 2 cos 2 = a 2 . (5) The lemniscate r 2 = a 2 cos 2 0. (6) The logarithmic 
spiral r = e a ^. (7) The spiral of Archimedes r = a<p. (8) The general 
spiral r = a<f> n . 

2. Derive the expression for R in Note 1. 



160 



DIFFERENTIAL CALCULUS. 



[Ch. X. 



103. Evolute of a curve. Corresponding to each point on a 
given curve there is a centre of curvature. The locus of the 
centres of curvature for all the points on the curve, is called 

the evolute of the curve. 
Thus, if AA 1 be the 
given curve and Q, 
@2> C 3 , -•', be respec- 
tively the centres of 
curvature for any 
points A 1} A 2 , A s , •••, 
on the given curve, 
the curve C X C 2 C Z is 
the evolute of AA V 

To find the equation 
of the evolute of the 
curve. Let the equa- 
tion of the given 
curve be 

2/ =/(*), (1) 

and let A(x, y) be any point on it. Let C be the centre of curva- 
ture for the point A, and denote C by (a, /?). Then [Art. 101, 

E 1- ( 2 )1 1 + AfcN. 




x — a 



dx 2 

1-f 



dy 
dx' 



y-P = - 



dry 

dx 2 



(2) 



(3) 



On the elimination of x and y from equations (1), (2), (3), there 
will appear an equation which is satisfied by a and j3, the coordi- 
nates of the point C. But A is any point on the given curve, and, 
accordingly, C is any of the centres of curvature for the points on 
AA X . Accordingly, the equation found as indicated is the equa- 
tion of the evolute. 

Note. The algebraic process of eliminating x and y from (1), (2), and 
(3) depends on the form of these equations. 



103, 104.] CIRCLE OF CURVATURE. 161 

EXAMPLES. 

1. Find the o volute of the parabola 

y 2 = 4px. (Fig. 48 a.) (1) 

Here by Ex. 3, Art. 101, a = 2p + 3 x ; (2) 

'=£■ (3) 

The elimination of x and y between equations (1), (2), (3), gives the 
equation of the evolute, viz. the semi-cubical parabola 

4(a-2p) 3 = 27j>/3 2 ; 

i.e. on using the ordinary notation for the coordinates, 

4(<c - 2 pY = 27 py 2 . 

2. Find the evolute of the ellipse b 2 x 2 + a 2 y 2 = a 2 b\ (Fig. 48 J.) (1) 
Here, by Ex. 4, Art. 101, a = / « 2 - b 2 \ 3 ^ ^ 

b 2 ' 



P 



=(^y. 



The elimination of x and y between equations (1), (2), (3), gives the equa- 
tion of the evolute, viz. : 

(aa)s + (6/3)1 = (a 2 - 6 2 )i, 

i. e. on using the ordinary notation for coordinates, 

(ax)i + (by)i = (a 2 - b 2 )h 

3. Find the evolute of the following curves : (1) the hyperbola b 2 x 2 — a 2 y 2 
= a 2 b 2 . (2) The equilateral hyperbola xy = a 2 . (3) The four-cusped hypo- 
cycloid £3 -f y 3" = a». 

4. Find both geometrically and analytically the evolute of a circle. 

5. Show that the evolute of a complete arch of a cycloid consists of the 
halves of an equal cycloid. [Suggestion ■ see Ex. 5 (8), Art. 101.] 

104. Properties of the evolute. The two most important proper- 
ties of the evolute of a curve are the following : 

(a) Tlie normal at any point of a given curve is a tangent to the 
evolute, and any tangent to the evolute is a normal to the given curve. 

(b) The length of an arc of an evolute, provided that the curva- 
ture varies continuously from point to point along this arc, is 
equal to the difference between the lengths of the two radii of curvature 
drawn from the given curve to the extremities of the arc. 



162 



DIFFERENTIAL CALCULUS. 



[Ch. X. 



Proof of (a). Let AA 1 (Fig. 47) be the given curve, and let its 
equation be y = f(x), aud let CG L be its evolute. Let C(«, /8) be 
the centre of curvature for any point A(x, y). 

The slope of the given curve at A is -^, and the slope of the 

7 n CLX 

evolute at C is -p • From Equations (2), Art. 101, on differentia- 
dot 



tion and reduction, 
da; " 



dxKdx 2 ) \_ \dx) _ 



da? 



dx 2 ] 



da dx { dx\dx z J \dxj Jaarj 

dx 



dx 2 ) 



From (1) and (2), and Art. 34 (3), 



da 



dx 



da 
dx 



doc 
dy 



(1) 



(2) 



(3) 



dx 



But — ^ is the slope of the normal at A (x, y). Hence, the 
normal at A and the tangent to the evolute at C coincide. 

F 

r 




'A x 



Fig. 48 a. 



Fig. 48 6. 



Note 1. Thus, in Fig. 47, AC is the radius of curvature for A on AA^ 
AC is normal to AA\ at A, and AC touches the evolute CC\ at C. In Figs. 
48 a, 48 6, PiOi, P 2 C 2 , are normal to the parabola and tangent to its evolute ; 
PC is normal to the ellipse and tangent to its evolute. 



104.] THE EVOLUTE. 163 

Note 2. On account of property (a) the evolute is sometimes defined as 
the envelope (see Art. 120) of the normals of the curve. See Art. 123 (Ex. 2 
and Notes 4, 5) and Art. 124, Ex. 1. Also see Echols, Calculus, Arts. 
106-108. 

Proof of (6). In Fig. 47 AA 1 is the given curve, CC X is its evo- 
lute, and C(ct, /?) is the centre of curvature corresponding to the 
point A(x, y). 

Let ds denote the differential of the arc of the evolute CC V 

Then, by Eq. (5), Art. 67 (c), 






dx * \<^/v dx 
.-. from (1) and (3) 



-v 



3 



1 + 



dy\ 2 dx \ dx 2 J 



<%(<&)* -[l+ftbi) 






\dxj 



dh/ 

dx 3 



dx) fdry^ 2 

dx\ 



(6) 



Differentiation of R in Art. 101, Eq. (1), gives 

rl 7? 

. — = the second member in (6). 
dx 

Hence ds = dR m ^ 

dx dx 

This means that at any point on the evolute CC X the rate of 
change of the length of the arc with respect to the abscissa x, is 
the same as the rate of change of the length of the radius of cur- 
vature at the corresponding point on AA X (Art. 26). It follows 
that on starting from two corresponding points (viz. a point on 
the curve and its centre of curvature) these lengths change by 
the same amount. Accordingly, 

the length of an arc of the evolute is equal to the difference between 
the lengths of the radii of curvature which touch this arc at its 
extremities ; or, in other words, the difference between the radii 
of curvature at two points on a curve is equal to the arc of the 
evolute intercepted between the centres of curvature of these points. 



164: DIFFERENTIAL CALCULUS. [Ch. X. 

Thus in Fig. 47, arc CC X = A X X - AC; arc <7 2 <7 3 = 4A - A 2 C 2 . 

Note 1. Property (&) is also shown in Art. 214. 

Note 2. Property (6) should not be applied thoughtlessly ; for in certain 
circumstances, for either the curve or its evolute, the property does not hold. 
Thus in the case of the curve ay 2 = cc 3 , the theorem is true only for points on 
the curve which are either both above the x-axis or both below. Again, in 
Fig. 48 a the theorem is true only for arcs of the evolute which are altogether 
above or altogether below the cc-axis. For instance, if (Fig. 48 a) P\C\ = 
P 2 C 2 , a reckless application of the theorem obtains the result 

arc C X SC 2 = P 2 C 2 - PiCi = 0, 
which is obviously absurd. 

Note 3. See Echols, Calculus, Art. 170 and Chap. XIV. 

Ex. 1. Show that the total length of the evolute of the ellipse whose 

4(Y/3 _ 7)3^ 

semi-axes are a and &, is — — '- • 

db 

Ex. 2. Show that the length of the evolute of the parabola y 2 = 4px that 
is intercepted by the parabola (i.e. 2 SB, Fig. 48 a) is 4p (3V3 — 1). 

105. Involutes of a curve. In Fig. 47 the curve CC Y is the 
evolute of the curve AA X . Suppose that a string is stretched 
tightly along the curve CC X and held taut in the position 
LC X G 2 C Z C, the portion LC Y thus being tangent to the evolute 
at C Jm Now, a point A x being taken in the string, let it be 
unwound from C\C. It follows from properties (a) and (6), 
Art. 104, that, as the string is unwound from the evolute C X C, A± 
will describe the curve A Y A. It is on account of this property 
that GC Y is called the evolute of AA V On the other hand, AA 1 
is called an involute of CQ. " An involute," because CC Y has an 
infinitely great number of involutes. For, when the string is 
unwound from the evolute C X C an involute will be traced out 
by each point like A 1 taken in the string LA X C X C 2 C Z . These 
involutes are parallel curves* ; for (1) they have the same normals, 
namely, the tangents of their common evolute, and (2) the dis- 
tance between any two of them along these normals is constant, 



* Two curves are said to be parallel when they have common normals 
always differing in length by the same amount. 



105.] EXAMPLES. 165 

namely, the distance between the two points originally taken on 
the string that is being unwound. Figure 47 shows three involutes 
of CC,. 

EXAMPLES. 

1. Construct several involutes of the evolute of the parabola whose latus 
rectum is 8 (besides the parabola itself). 

2. Construct several involutes of the evolute of the ellipse whose axes 
are 9 and 25. 

3. Given a cycloid, construct the involute that is traced out by the point 
at the vertex in the course of "the unwinding." 

4. Given a circle, construct the involute that is traced out by any point 
on the circle in the course of "the unwinding.'" (In the case of a circle 
all such involutes are identically equal. Accordingly, such an involute is 
usually termed "the involute of the circle.") 

5. Construct several involutes of an ellipse, and several involutes of a 
parabola. 



CHAPTER XI. 

ROLLE'S THEOREM. THEOREMS OF MEAN VALUE. 
APPROXIMATE SOLUTION OF EQUATIONS. 

106. In this chapter two theorems of great value in the cal- 
culus are discussed, viz. Rolle's Theorem and the Theorem of 
Mean Value. The truth of the latter theorem is made manifest 
in a geometrical or intuitional manner in Art. 108 ; in Art. 110 it 
is deduced from Rolle's Theorem. Since there are several mean- 
value theorems in the calculus, the Theorem in Arts. 108, 110, 111 
may be called the First Mean-value Theorem. Another mean- 
value theorem is given in The Integral Calculus, Art. 213. 
Rolle's theorem and the first mean-value theorem are funda- 
mental, and play a highly important part in the modern rigorous 
exposition of the calculus. Two other mean-value theorems are 
deduced in Arts. 112, 113. The theorem in Art. 113 is required 
in Chapter XVI. An application of the mean-value theorem is 
made to the approximate solution of equations in Art. 109. 

107. Rolle's Theorem. 

Note 1. Progressive and regressive derivative. In Art. 22 the derivative 

lim te+M^m. (i) 

Ax 
The process of evaluating (1) is equivalent 
to the geometrical process of revolving the 
chord PQ of tl e curve y = /(x) about P until 
Q coincides with P, and thus PQ becomes the 
tangent PT. If in this eurve a chord PB be 
drawn, and BP be revolved about P until B 
coincides with P, then BP will finally take 
the position PT. The slope of the tangent 
obtained by thus revolving BP is evidently 

Km^ f(x) -- f(x ~ Ax) ; i.e. ih^ fcMda , (2) 

Ax — Ax 

166 



of /(x) was defined as 




106, 107.] 



BOLLE S THEOREM. 



167 



It is customary to call (1) the progressive derivative, and (2) the regressive 
derivative.* In general these derivatives are equal ; that is, in general the 
tangent on the representative curve is the same, whether the secant which is 
revolved until it assumes a tangential position be drawn forward or backward 
from the point under consideration. In some cases, however, these deriva- 
tives are not equal ; such a case is represented at P on Fig. 51 c, where the 
two revolving secants give two different tangents. In such a case the deriva- 
tive is discontinuous at P, for its value suddenly changes from the slope of 
TP to the slope of LP. 

Theorem. If a function f (x) and its derivative f(x) are continu- 
ous/or all values of x between a and b, and if f(a)=f(b), then 
/'(.?) = for at least one value ofx between a and b. 

Following is a geometrical proof f and representation of this 
theorem. Let the curve MN (Figs. 50 a, b, c) represent the 
function /(a). 

At M and N let x = a and x = b respectively. Since the ordi- 
nates AM and BX are equal, it is evident that there must be at 
least one point between M and JV where the function ceases to 
increase and begins to decrease, or ceases to decrease and begins 
to increase. There may be several such points, as in Fig. 50 c.$ 
But at such a point, for instance P, or P 1} or P 2 , or P 3 , the value 
of the first derivative, which is continuous by hypothesis, must 
be zero. 





P 




\^^ 






Pt 








F 




F 










F 


Am 




N 


V 




P". 






\ 






A 


t 


M 






V 






A 


B 






A 




Pa 








B 








A 




Pi 


B 




O 


'a 








XO 


u 




XO 


Xti 




jj 




X 
























































Fig. 50 a. 



Fig. 50 6. 



Fig. 50 c. 



*They are also called right- and left-hand derivatives. 

f An analytical discussion will be found in the collateral reading suggested 
in Note 3, Art. 108. 

% Here functions having only a finite number of oscillations between M 
and N are dealt with. On the relation between RohVs theorem and func- 
tions having an infinite number of oscillations between M and iV, see Pier- 
pont, Functions of Peal Variables, Vol. I., Arts. 394-396. 



168 



DIFFERENTIAL CALCULUS. 



[Ch. XL 



A special case of this theorem is that in which f(a) = and 
f(b)=0. The student may construct the figure for himself by 
merely moving OX to the position MN. The statement of the 
theorem for this case is usually taken as the general statement 
of the theorem. It is as follows : 

Rollers Theorem (second statement) : 

If f{x) is zero ivhen x = a and when x = b, and f(x) and its de- 
rivative fix) are continuous for all values of x between a and b, 
then f(x) will be zero for at least one value of x between a and b. 

Note 2. The necessity of the condition relating to continuity is evident 
from Figs. 51 a, b, c, d. 




Fig. 51 a. 



Fig. 51 5. 



Fig. 51 c. 



Fig. 51 d. 



For a value of x between x = a and x = b : in Fig. 51 «, f(x) is infinite ; 
in Fig. 51 6, f(x) is discontinuous ; in Fig. 51 c, f'(x) is discontinuous ; in 
Fig. 51 d,f'(x) is infinite. 

Note 3. The theorem does not necessarily fail if f'(x) is infinitely great 
for some value of x between a and b. For instance, if there is a vertical 
tangent at a point of inflexion between P 2 and P 3 or at a point between P 3 
and P 4 , Fig. 50 c (tangents as in Fig. 26 6) , the theorem still holds true. 

Note 4. Algebraic application of Rolle's Theorem. 

An important application of Rollers Theorem may be made to the 
theory of equations. According to the theorem, geometrically, 




7U)=o f(x)=o 

Fig. 52 a. 



Fig. 52 b. 



107, 108.] 



THEOBEM OF MEAN VALUE. 



169 



the slope of a curve y — f(x) is zero once at least, between the 
points where the curve crosses the se-axis. Hence, at least one 
real root of the equation f(jx)= lies between any two real roots 
of the equation f{x) = 0. In the theory of equations this is 
called Eolle's Theorem, after Michel Rolle (1652-1719). 

Note 5. According to this principle r real roots of an equation f(x) = 
have at least (r — 1) roots of f(x) = between them. Now., if the r roots 
coalesce and thus make an r-tuple root, the (r — 1) roots must also coalesce 
and thus make an (r — l)-tuple root oif(x) = 0. (See Art. 66a.) 

Ex. Verify Eolle's Theorem in each of the following equations /(x) = ; 
also sketch the curve y =f(x): 

(1) x 2 + x - 6 = ; (2) as* + 2 x* - 5 x - 6 = 0. 

108. Theorem of mean value. If a function f(x) and its derivative 
f(x) are continuous for cdl values of x from x = a to x = b, then 
there is at least one value of x, say x x , between a and b such that 

b — a 

i.e. such that/(6)=/(a) + (& — a)/'^). 

Following is a geometrical proof* and explanation of this theorem. 

Let the curve MN (Fig. 53 a or Fig. 53 b) represent the func- 
tion f(x). Draw the ordinates AP and BQ at A and B, where 



/'W; 




Fig. 53 b. 



x = a and x = b respectively. Draw PQ and draw PR parallel to 
OX. Then AP = f(a), BQ=f<b), 



* For an analytical deduction of the theorem of mean value from Eolle's 
Theorem, see Art. 110. 



170 DIFFERENTIAL CALCULUS. [Ch. XL 

Hence RQ = f(b) - /(a), 

and t ^ BPq= m = m-m. 

* PR b-a 

Now the chord PQ and the tangent ST drawn at some point V 
(or V\ and V 2 ) between P and Q evidently must be parallel. At 
V let x = Xu x 1 thus being between a and b; then tan RPQ=f'(x } ). 

Hence ^iff'W w- (1) 

Since x 1 is between a and b, x i = a + 6(b — a), in which 6 
denotes some number between and 1 (i.e. O<0<1). Accord- 
ingly, theorem (1) may be expressed 

f(b)=f(a) + (b - a)f[a + 6(b - a)]. (2) 

If b — a = h, then b = a + 7i, and (2) is written 

Aa + h) = f(a) + hf'(a + 9ft). (3) 

Eesult (3) has important applications. It is very useful for 
finding an approximate value of /(a + 7i) when f(x) f a, and h, are 
given. A closer approximation to the value of f(a-\-h) can be 
found by Taylor's formula, Art. 150. 

Note 1. The necessity for the condition relating to continuity can be 
made evident by figures similar to Figs. 51 a, 6, c, d. 

Note 2. The remark in Note 3, Art. 107, applies also to the mean-value 
theorem. In cases, however, in which f'(x) may be infinite for values of x 
between a and b, X\ in (1) must be such that/'(o;i) is finite. 

Note 3. References for collateral reading on Bolle's theorem and the 
theorem of mean value: McMahon and Snyder, Diff. Cal., Arts. 59, 66; 
Lamb, Calculus, Arts. 48, 49, 56; Gibson, Calculus, §§ 72, 73; Harnack, 
Calculus, Art. 22 ; Echols, Calculus, Chap. V. The last mentioned text has 
a particularly full and valuable account of these theorems. Also see Pierpont, 
Functions of Real Variables, Vol. I., Arts. 393-404 ; Goursat-Hedrick, Math- 
ematical Analysis, Vol. I., Arts. 7, 8 ; Osgood, Calculus, Chap. XL 

EXAMPLES. 

1. Find by relation (3) an approximate value of sin 32° 20' taking a = 32° : 
(1) putting 6 = 0, (2) putting 6 = 1; and compare the calculated results 
with that given in the tables. 



108,109.] APPROXIMATE SOLUTION OF EQUATIONS. 171 

2. If f(x)= 2x 2 — x + 5, find what d must be in order that relation (3) be 
satisfied : (1) when a = 3 and h = 1 ; (2) when a = 10 and /i = 2. 

3. Show that for any quadratic function, say f(x) = lx 2 + mx + w, 
/(a + h) will be obtained by putting = ^in relation (3). What geometrical 
property of the parabola corresponds to this ? {Deduce the value of 6.) 

4. If/(z)= a; 3 , find what must be in order that relation (3) be satisfied 
when a = 3 and ft = 1. What problem in connection with the cubical 
parabola y = x 3 is the correlative of this? 

109. Approximate solution of equations. The real roots of an 
equation can generally be found to as close an approximation as 
one pleases by the help of the calculus. 

Let f(x)=0 (1) 

be the equation. Suppose that an approximate value of a root of 
(1) has been found, by substitution or otherwise, and suppose this 
value, say the nearest integral number in the root, is a. 
Suppose the corresponding root of (1) is a -j- h. 

Then f(a + h) = 0. (2) 

But, by Art. 108, result (3), 

f(a + h)=f(a) + hf'(a + 6h), [-1<0<1]. (3) 
From (2) and (3), 

f(a) + hf'(a + 6h)=:0. 
An approximate value of h, say \, may be found by taking 
6 = 1, and putting f(a) + hJ , {a) = ^ (4) 

This gives / ll = _iM. 

Accordingly, a second (and, in general, a closer) approximation 
to a root of (1) is f( n \ 

*-?§)' ^ 

On starting with this value as an approximate value of the root, 
and again proceeding in a similar way, a still closer approxima- 
tion to the root may be found. This process may be repeated as 
often as may be deemed necessary * 

* This method of finding an approximate solution of an equation is called 
Newton's method. 



172 DIFFERENTIAL CALCULUS. [Ch. XL 



EXAMPLES. 

1. Find approximately a root of the equation 

x 3 + 2 x - 19 = 0. (6) 

Here /(2) = — 7, and /(3) = + 14. Accordingly, at least one root of the 

equation lies between 2 and 3.* Since 2 is evidently nearer the value of 

the root than 3 is,t let the number 2 be chosen as the first approximation to 

the root. 

In this example, /(x) = x 3 + 2 x — 19. 

Hence /'(x)= 3x 2 + 2, 

and /(2) = -7, /'(2)=14. 

7 

.-. by (5), a closer approximation to the root than 2 is 2 — • i.e. 2.5. 

14 
Now taking 2.5 as an approximate value of a root of (6), 

a closer approximation = 2.5 - ISML = 2.5 - M^ = 2.5 - .07 = 2.43. 

/'(2.5) 20.75 

Using 2.43 as an approximate value, 

a closer approximation = 2.43 - Z( 2 - 43 ) = 2.43 - - 208907 

/'(2.43) 19.7147 

= 2- 43 -.0106 = 2.4194. 

2. Find a root of x 3 - x 2 - 2 = 0. 

Substitution gives/(l) = — 2, /(2) = + 2. Accordingly, a root lies be- 
tween 1 and 2. 

Here f( x ) = x 3 — x 2 — 2. 

.-./'(x)=3x 2 -2x. 
It will be found better to take 2 for a first approximation to the root. 

A second approximation = 2 — ^ ^ = 2 — | = 1.75. 



A third approximation = 1.75 — ^^ — ^ = 1.7£ 



.296875 



/' (1,75) 5.6875 

= 1.75- .05219 
= 1.698. 
If 1 be taken as an approximation instead of 2, the process for finding the 
next approximation gives 3, which is farther from the root than 1 or 2. Thus : 

second approximation = 1 — J-±-)- = 1 — ^— = 3. 
./'(I) 1 

An explanation of this result is given in Note 1. 

* In this case, when x changes from 2 to 3, /(x) changes from — 7 to + 14. 
Now /(x) is a continuous function of x. Accordingly, f(x) must pass through 
zero once at least when it is changing from the negative value (— 7) to the 
positive value (+ 14). t For — 7 is nearer zero than + 14 is. 



109.] 



EXAMPLES. 



173 



Ex. Taking 3 as an approximation to a root of the above equation, 
derive successive approximations therefrom. 

Note 1. Suppose x = Xi is taken as an approximation to a root of the 
equation f(x) = 0. 

Consider the equation of the tangent to the curve 

y=f(x) 

at the point whose abscissa is sci, say the point Oi, y{). Here y± =/(xi). 
The equation of this tangent is 

y-Vi =f'(x 1 )(x-x h . 

On proceeding as shown in analytic geometry, it is found that this line 
crosses the x-axis where 



Xl 



Accordingly [see (5)], the above method of finding a second approxima- 
tion to a root of f(x) — 0, on starting with an approximation xi, is practically 
the same as finding where the tangent at 
(&ii yi) on the curve y=f(x) intersects 
the x-axis. 

When the abscissa of this intersection 
is outside the limits between which the 
root is known to lie, the method fails. 
This is shown in Fig. 54, which illustrates 
Ex. 2. 



SL is the curve 



X 3_£2_2. 



At A, x = 1 ; at B, x = 2. The curve 
crosses the x-axis at D, between A and B. 
The abscissa OB represents the real root of 
the equation 

x 3 - x 2 - 2 = 0. 

On proceeding as shown in Art. 61, it 
will be found that : 





Y 




A 


L 

\\ 
l\ 

h 

\\b 


T 







w 




V 


/C X 


s/ 


/ 











Fig. 54. 



the tangent PT, at P where x = 1, crosses the x-axis at C where x — 3 ; 
the tangent QR, at Q where x = 2, crosses the x-axis at V where x — 1.75. 

Note 2. Another method of finding an approximate solution, when the 
equation is- algebraic, is LTorner''s* method. This is described in text-books 
on algebra. 



* Also see pages 247, 256. 



174 DIFFERENTIAL CALCULUS. [Ch. XI. 

Yet another method of finding an approximate solution of an equation is 

the graphical method. This is described in various text-books. Thus, to 

solve the equation 

ajs _ X 2 _ 2 = 0, 

carefully plot the curves y = x 3 , 

y=x 2 + 2, 
and obtain the abscissa of their point of intersection. At this point 
x 3 = x 2 + 2, i.e. x s - x 2 - 2 = 0. 

Another example : to solve the equation 

x = 3 sin x, 
carefully plot the curves y = - , 

2/ = sin x, 
and obtain the abscissa of their point of intersection. At this point - = sin x, 
i.e. x = 3 sinx. 

Ex. Solve these examples by the graphical method. 

Note 3. In connection with this article, see Osgood, Calculus, Chap. XX., 
Arts. 1-5. 

EXAMPLES. 
Find approximate solutions of the following equations : 

1. x s - 12 x + 6 = 0. 6. x 3 + 4 x 2 + x + 1 = 0. 

2. x 3 + x 2 - 10 x + 9 = 0. 7. a? = 6. 

3. x* - 12 x 2 + 12 x- 3 = 0. 8. x 3 -4x-2 = 0. 

4. x 3 + 3 x - 20 = 0. 9. 2 x 3 + x 2 - 15 x - 59 = 0. 

5. e*(l + x 2 ) = 40. 10. x? - 6 x 2 + 3 x + 5 = 0. 

11. x 3 -3x-4 =0. 

110. Theorem of mean value derived from Rolle's Theorem. Let 

f(x) and its first derivative f'(x) be continuous in the interval 
from x = a to x = b. 

O a X\ 



Fig. 55 * 



Consider the quantity Q which represents the difference-quotient 

l the equation, /(&)-/(«) = q. (1) 

b — a 

From (1), f(b) - f(a) - (b - a) Q = 0. (2) 

* In connection with Figs. 55-60, see Art. 15 a and Fig. 5, footnote. 



110, 111.] THEOREM OF MEAN VALUE. 175 

Let F(x) denote the function formed by replacing b by x in the 
first member of (2) ; that is, let 

F(x) = f(x)-f(a)-(x-a)Q. (3) 

Then, FQ>) = f(b) - f(a) -(b-a)Q = 0, by (2) ; (4) 

also, F(a)=f(a)—f(a)—(a — a)Q = 0, identically. (5) 

Now f(x) and f'(x) by hypothesis are continuous in the interval 
(a, 6); also (x — a)Q is a continuous function, and its derivative 
Q is a constant. Accordingly, from these facts and equation (3) 
it follows that 
F(x) and its derivative F'(x) are continuous in the interval (a, b). 

Also, F(x) is zero when x = a and when x = b. [Eqs. (4), (5).] 

Thus the conditions of Rolle's Theorem (second statement) are 
satisfied by F(x), and therefore 
F'(x) will be zero for at least one value of x, x 1 say, between a 

and b ; 

that is F'(x 1 ) = 0, in which a < a\ < b (see Fig. 55). (6) 

From (3), on differentiation, F f (x) = f'(x)- Q. (7) 

.-. on substitution of x ± in (7), F t (x i ) = f'(x^)— Q; (8) 

whence by (6) and (8), Q =f'(x 1 ), a < x l < b. (9) 

Substitution from (9) in (1) gives 

(&l=^l = f( Xl - ) ,a<x 1 <b. (10) 

111. Another form for the theorem of mean value. 

From Art. 110 (10), /(&) = /(a) + (b - a)f(&), a < x 1 < &. (1) 

Suppose b — a = h. (See Fig. 55.) 

Then & = a + h ; 

and, since a^ is between a and 6, 

ajj = a + 07i, 
in which denotes a proper fraction, i.e. < < 1. 

Then (1) can be written 

Ka + h) = f(a) + hf(a + Qh), O<0<1. 

(See Art. 108, Eq. (3) and on.) 



176 DIFFERENTIAL CALCULUS. [Oh. XI. 

112. Second theorem of mean value. If a /miction of fix) and 
its first and second derivatives, f'(x), f"(x), are continuous for all 
values of x from x — a to x = b, then there is at least one value of x, 
say x 2 , between a and b such that 

f(b) = f(a) + (b- a)f(a) + J (6 - affix,). 

The proof proceeds on lines similar to those in Art. 110. 

4 £ P ■ ? : — 

Fig. 56. 

Consider the quantity R in the equation 

f(b)-f(a)-(b-a)f'(a)-i(b-ayR = 0. (1) 

Let F(x) denote the function formed by replacing b by x in the 
first member of (1) ; that is, let 

F{x) =f(x) - f(a) -(x- a)f(a) -\{x- afR. (2) 

Then F(a) = 0, identically ; and F(b) = 0, by (1). 

Also, it follows from equation (2) and the hypothesis of the 
continuity of f(x) and f'(x) that F(x) and F'(x) are continuous 
in the interval (a, b). Thus the conditions of Eolle's theorem 
are satisfied by F(x), and therefore 
F'(x) will be zero for at least one value of x, x x say, between 

a and b ; 
that is F'(x^ = 0, in which a < x l < b. (3) 

From (2), on differentiation, 

F'(x) = f'(x) - f\a) -{x- a)R. (4) 

Hence, from (3) and the substitution of x x in (4), 

F\x x ) = f( Xl ) - f'(a) - (x, -a)R = 0. (6) 

Also, from (4), F'(a) = f(a) - f\a) - (a - a)R = 0. (7) 

Further, it follows from equation (4), and the continuity of 
f(x), f"(x) and F'(x), that F"(x) is continuous in the interval 
(a, b). Thus the conditions of Rolle's theorem are satisfied by 
F'ix) in the interval (a, x^, and therefore 



112, 113.] EXTENDED THEOREMS OF MEAN VALUE. 177 

F"(x) will be zero for at least one value of x, x 2 say, between a 
and x lf and thus between a and b ; that is 

F"(x 2 ) = Q, a<x 2 <b. (8) 

From (4), on differentiation, F"(x) = f"(x)- B-, (9) 

whence, on substitution of x 2 , F" (x 2 ) = f n (x 2 ) — B. (10) 

From (10), by (8), B = f"(x 2 ), a<x,<b. (11) 

Substitution of this value of B in (1), and transposition, give 

iW =/(«)+(»- «)/"(«) + |(&-«) 2 /"(^ 2 ), «<^ 2 <&. (12) 

Another form of theorem (12). 

On denoting the interval 6 — a by 7i, and proceeding as in 
Art. Ill, relation (12) will take the form 

f(a + h) = f(a)+ hf'(a)+\h*f*(a + 9^), O<0 1 <1. (13) 

113. Extended theorem of mean value. A. First method. Sup- 
pose that f(x) and its first three derivatives f'(x), f"(x), f'"(x), 
are continuous in the interval from x = a to a; = b. By the same 
method as that used in Art. 112 a number S can be considered 
which satisfies the equation 

f(P) -/(«) - (P - «)/'(«) - i (6 - «) 2 /"(«) - 273 ( & - «)* a = °- W 

It will be found that S =f'"(x 3 ), in which x 3 is a value of a; 
between a and 6. 

Substitution of this value of S in (1) and transposition give 

/(&) =f(a) + (b - a)f(a) + ^|)! / » (a) + ^zg/-^), ( 2 ) 

in which a < # 3 < 6. 

Suppose that f(x) and its first n derivatives are continuous in the 
interval from x = a to x=b. By following this method succes- 
sively there will at last be obtained the extended theorem of mean 
value : 
/(&) =/(«) + (& -»)/'(«)+ (& -^ f»(a)+ (6 ~ ! a) V // W+ - 

+ ^=f^/<»>G*„), (3) 

in which a<oc n <b. 



178 DIFFERENTIAL CALCULUS. [Ch. XI. 



x n b 
H 1- 



Since x n is between a and b, x n = a + 
FlG 57 6 (b — a), in which < 6 < 1. 

On denoting b — a by ft, and proceed- 
ing as in Art. Ill, result (3) will take the form 

f{a + ft) = /(«) + hf'{a)+ !?/ '(«) + |?/'"(a)+ - 

+ ^ n) (« + e,^), (4) 

in which n is a fraction between and 1, i.e. 0<6 n <l. 

B. Second method. Theorem (3) can also be obtained by a 
single application of Eolle's theorem. 

Let f(x) and its first n derivatives be continuous in the inter- 
val from x = a to x = b. Consider R n in the equation 

JQ>) -/(<*) - - «)/'(«) - i- (6 - «) 2 /"(«) 

- ( (»"i a i)T /( "" 1)(a) ~ (6 ~ a) " R " = a (5) 

Let i^V) denote the function formed by replacing a by a? in the 
first member of (5) ; that is, let 

F(x) =/(&) -f{x) -(b- x)f(x) - 1 (b - xff \x) +.- 

(n. — 1) ! 

Since /(a?) and its first ?i derivatives are continuous in the in- 
terval from x = a to x = b, it follows from equation (6) that 

F(x) and i^'(x) are continuous in this interval. 

Also, F(a) = 0, by (5) ; and F(b) = 0, identically. 

Thus the conditions of Eolle's theorem are satisfied by F(x), 
and therefore F'(x) will be zero for at least one value of x, x n say, 
between a and b ; that is 

F'(x n )=Q, a<x n <b. (7) 

From (6), on differentiation and reduction, 

F\x)=- ( f~^*~> >(aO + n(b - xy-*B n ; 
(n-l)l 



113.] EXTENDED THEOREMS OF MEAN VALUE. 179 

whence, on substitution of x n for x, 

F\x n ) = - ( (~yrV (w) (^«) + nib - x n y-'R n . (8) 

From (8) it follows, bv virtue of (7), that 

Jt = i/»«). (9) 

Substitution of this value of R n in (5) and transposition give 
formula (3) above. 

N. B. Another theorem of mean value commonly called the 
Generalized Theorem of Mean Value is given in Art. 116, Chap. XIII., 
where it is needed for immediate application. 



CHAPTER XII. 

INDETERMINATE FORMS. 

114. Indeterminate Forms. Functions sometimes take peculiar 

x 2 — 4 
forms. For instance, — , 

x — 2 

when x = 2, 

has the meaningless form -• 



Special instances in which this form presents itself have been 

considered in preceding articles ; e.g. — " and — in Chap. I., and 

Ax At 

in Arts. 22, 24, 25; ^, — , in Exs. 7, 8, Art. 14. 
6 

When x = the function x cot x has the form • oo ; 

when x = - the function (tan#) cosx has the form go . 
2 v ; 

Cases like these, and others to be mentioned, require further 
special examination. These peculiar forms are called indetermi- 
nate forms. They are also called illusory forms, The object of 
this chapter is to show the calculus method of giving a definite, 
a determinate, value to a so-called indeterminate form. 

There are various other methods, which are sometimes simpler 
than the method of the calculus, for " evaluating " functions 
when they take illusory forms.* All the methods, however, start 

* " In the present chapter we propose to deal specially with these critical 
cases of algebraical operation, to which the generic name of "Indeterminate 
Forms " has been given. The snbject is one of the highest importance, inas- 
much as it forms the basis of two of the most extensive branches of modern 
mathematics — namely, the Differential Calculus. and the Theory of Infinite 
Series (including from one point of view the Integral Calculus). It is too 

180 



114,115.] INDETERMINATE FORMS. 181 

with the same fundamental principle, or rather with the same 
definition, concerning what is to be taken as the value (sometimes 
called 'the true value ') of an indeterminate form. The princi- 
ple on which a value is assigned is illustrated in Arts. 117, 118. 
Briefly stated, the principle is this : 

Siqypose a function f(x) takes an indeterminate form when 

x = a. 
Tlie value of /(«) is defined as 
the limit* of the value of /(as) when x approaches a. 

Note 1. Definition A really takes that value for/(x) which makes the 
function /(x) continuous when x = a. This may be indicated arithmetically 

in the case of the function ' ( ~ • For, when 
ar-2 

x takes the values 1, 1 • 5. 1 ■ 7, 1 • 9, 2, 2 • 1, 2 • 2, 2-3, ••• successively, the 
function takes the values 3, 3 • 5, 3 • 7, 3 • 9, 4, 4 • 1, 4 • 2, 4 • 3, ••• successively. 
The calculus method for obtaining the value 4 for the function when x = 2, 
is shown in Art. 117, Ex. 1. 

115. Classification of indeterminate forms. The following seven 
cases of indeterminate forms occur in elementary mathematics. 

/1N sin x -, A 

(1) q, e.g. 3-—, when x = 0. 

/f) s ac log a* , 

( 2 ) — ; e Q- — jj— > when »=oo. 

(3) qc — oc ; e.g. sec x — tan x, when x = — • 

(4) Ox; e.g. ( ^ — x ] tan x, when x = ^- 



much the habit in English courses to postpone the thorough discussion of 
indeterminate forms until the student has mastered the notation of the dif- 
ferential calculus. This, for several reasons, is a mistake. In the first place, 
the definition of a differential coefficient involves the evaluation of an inde- 
terminate form ; and no one can make intelligent applications of the differ- 
ential calculus who is not familiar beforehand with the notion of a limit, 
Again, the methods of the differential calculus for evaluating indeterminate 
forms are often less effective than the more elementary methods which we 
shall discuss below, and are always more powerful in combination with them." 
Chrystal, Algebra. Part II., Chap. XXV., § 1. * If there is such a limit. 



182 DIFFERENTIAL CALCULUS. [Ch. XII. 

(5) l x ; e.g. (1 +- ] , when x = cc . 

(6) 0°; e.g. x x , when x = 0. 

(7) ooO. e .g, (cot#) sina: , when x = 0. 

The l evaluation ' of forms (3)-(7) can be reduced to the evalua- 
tion of either (1) or (2). 

In this book the method of the calculus for evaluating forms 
(1) and (2) is made to depend upon an important mean-value 
theorem — the generalised theorem of mean value. This theorem is 
given in the next article. 

116. Generalized theorem of mean value. Iff(x), F(x), and their 
derivatives f (x), F'(x), are continuous in the interval from x = a to 
x = b, and if F'(x) is not zero when x is between a and b, then 



(1) 



F(b)-F{a) F f (a^y 

in which a < x\ < b, a ^i & 

Fig. 58. 
Consider the function <f>(x) in the equation 

*<*> = F(b)-F(a) j F(X) ~ F(a) S " i/(X) ~ /(a) 5 • (2) 

Since f(x), F(x), fix), F'(x) are continuous in the interval (a, b), 
it is apparent on an inspection of (2) that the function <f>(x) and 
its derivative <f>'(x) are continuous in this interval. 

Also, from (2), <f>(a) = f identically; and <£(&) = 0, identically. 

Thus <J>(x) satisfies the conditions of Rolle's theorem. 

.'. <j>'(x) will be zero for at least one value of x, x x say, between a 

and b ; that is ^'(a^) = 0, in which a < x 1 < b. (3) 

From (2), on differentiation, 

*' (x) = FS)-F$) F ' {X) - f(X) ' ^ 

whence, on substitution of a\ for x, 



116, 117.] IXDETERMIXATE FORMS. 183 

From (5) it follows, by virtue of (3), that 

F(b) - JF(a) " F 7 ^) ' m * I11C11 a<x l< °' W 

117. Evaluation of functions when they take the form - • feefer- 

y 

ring to definition A, Art. 114, the determination of the limit 

mentioned there is called the evaluation of the function. 

Suppose f{x) and F(x) both vanish when x = a; that is, 

suppose f(a) = and F(a) = 0. (1) 

According to definition A } Art. 114, 

ralue of ^ is defined as H«^*>|. (2) 

Siqypose that a is finite. 

In the generalised theorem of mean value, Art. 116, Eq. 6, 
substitute x for b. 

Here x and x 1 must be such that 



a<x<^b and a<x x <x. £_£ ? ? 

Fig. 59. 
Then the theorem takes the form 

f(*)-fto =£S£L,a <Xl< x. (3) 

Since f(a) == and F(a) = 0, this becomes 

/M=/M «<.,<«. ( 4) 

Xow let as approach the limit a. Then, since a^ lies between a 
and x, x± must also approach the same limit a, and x and % must 
reach the limit a together. 

••■*-^-'--*g&-S& » 



184 DIFFERENTIAL CALCULUS. [Ch. XIL 

It sometimes happens that f(a) and F'(a) are both zero. When 

this is the case, the application of the same reasoning and process 

fix) 
to the function \ lV } when x approaches a, leads to the result 
F (x) 

value of lM = h&. ( 6 ) 

F'(a) F"(a) K } 

If the second member of (6) also has the same indeterminate 
form, the fraction formed by the third derivatives is required; 
and so on. It thus becomes evident that : 

If , for x = a,f(x) and F(x) and all their derivatives up to and 
including their nth derivatives, are zero, while f {n+l) (a) and F (n+1) (a) 
are not both zero, then 

theyalueof^ = ^ +1 ; (a) . (7) 

F{a) F(n+V(a) 

Result (7) may also be expressed thus : 

If a is infinite, substitute - for x and evaluate for z = 0. 

z 

It can be shown that this is practically the same as to put 
a = oo in relations (5) or (7). 

Note. In virtue of definition A, Art. 114, the following expressions may- 
be regarded as synonymous in the case of a function f(x), which takes an 
indeterminate form when x = a ; viz. 

" find the value of f(x) when x = a ; " " evaluate f(x) when x = a ;" 
" find the limit of f(x) when x approaches a " {i.e. "find lim^—afix) "). 

EXAMPLES. 

1. Evaluate ^ ~~ 4 when x = 2. (See Art. 114, Note 1.) 

x-2 v ' ; 

Valuer ^lA = value xi2 D ^ ~ ^ = value x = 2 — = 4. 

x — 2 D(x — 2) 1 

2. Evaluate (x — sinx) -f- a; 3 when x = 0. In this case, 

,■ x — sinx i. 1 — cosx* ,. sinx* ,. cosx 1 

hm^o = nm x ^o — — = hm^ — ; = hm^— — = - • 

x z 8x 2 bx 5 6 

* Which is in the form 0-fO. 



117. 118.] INDETERMINATE FORMS. 185 

Xote. The labour of evaluating /(a) -f- 0(a) may be lightened in the fol- 
lowing cases : 

{a) If, in the course of the reduction a factor, say ^(x), appears in both 
the numerator and the denominator, this common factor may be cancelled. 

(6) If at any stage during the process of evaluation a factor, say ^(x), 
appears only in the numerator or only in the denominator, and ^(«) is not 
zero, the value of ^(«) may be substituted immediately for rp(x). This will 
lessen the labor in the succeeding differentiations. 

3. Evaluate the following : (1) a * - h \ when x = 0; (2) sm " lx , when 

x x 

x = o ; (3) x ' 1 ~ an , when x = a j (4) e * - e ~* , when x = ; (5) 1 ~ cos z 

w x- a ' K sinx ' • z* 

when z = 0. 

4. Find the following : 

(1) lim^o^- 5 ) 2sina; ; (2) lim^ 2 ^- 5 ) 21 °g^-^ ; 

x sin (x — 2) 

/ox «„, e* + e~ J + 2 cos x - 4 , r± . ,. tan x — sin x . 

(o; nmx^o — ; (-4) lim X io : 5 

x 4 x — sin x 



(5) lim^o 



1 — cos x 



cosx sin- 5 X 

[Answers: Exs. 3. log-, 1, na n \ 2, |; Exs. 4. 25, -9, i, 3, |.] 

& 

118. Evaluation of functions when they take the form |g. Sup- 
pose f(x) and i 7 ^') are both infinite when x = a; that is, suppose 

/(a) = ao and F(a) = ao. 

Let the limiting valne of 

be required. 

Suppose that a is finite. Suppose that the conditions for the 
generalised theorem of mean value, Art. 116, are satisfied in an 
interval (x, b), in which x is some number such that 

a < x < b. 

For the interval (x, b) then, Theorem (6), Art. 116, has the form, 
f(b)-f(x) _f(x 1 ) (1) 

F(b)-F( X ) F\ Xl y w 

q x X\ q 

in which a < a; < a^ < 6. Fig. 60. 



186 DIFFERENTIAL CALCULUS. [Ch. XII. 

On changing signs and multiplying up, (1) becomes 

f(x)~f{h) = £&L\F(x)-F(b)\. (2 ) 

It is also supposed that F'(xj) is not zero, in the interval (a, b). 
On division of the members of (2) by F(x), 

/(«0 /(6) = /'fe) U F(b))_ (S) 

F{x) F(x) F'fa) 1 F(x) J 

Now let x approach a as a limit. Then, since 
f@L = and ^i = (because F(a)= oo and/(6) and F(b)=f= oo), 
equation (3) takes the form 

F{a) F'(xd V 

The first member in (4) has the form — , and the x x in the sec- 

00 

ond member is any number in the interval (a, b). The value ob- 
tained for the second member by letting x 1 approach a as a limit, 
is taken as the value of the first member ; that is 

value of £& = lim x „ a ^i; 
F(a) * F\x x )' 

value of >W = lim^^M = ^^. (5) * 

F(a) F'{x) F'(a) K J 

If -- ^ '- is also indeterminate in form, similar reasoning to 

F\a) & 

that in Art. 117 leads to the same general result (6) of that arti- 
cle. If a is infinite, the remarks made in Art. 117 for the same 
condition apply. 

It thus appears that the illusory forms in Arts. 117, 118, both 
are evaluated by the same process in the calculus. 



* For more rigorous derivations of the fact that the second member of (5) 

is the limiting value of • ■ ^ when x = a, see Gibson, Calculus, pages 420, 
F(x) 

421 ; Pierpont, Functions of Real Variables, Vol. L, Art. 452. 



118, 119.] INDETERMINATE FORMS. 187 

EXAMPLES. 

1. Evaluate x = when x = oo . (See Art. 8, Note 2.) 

log a: 

x 1 

linij^o — — = linij.^0 - = lim x= b3o x =oo . 
log x 1 

x 

2. Evaluate—, — , — , when a; = oo . 

e x e x e x 

3. Eind: (l)lim^ 1 ^; (2) Hm^- J~*- ; (3) lmw 1 -^*. 

cotx 2 sec 3 x 2 tanx 

[Answers: Exs. 2. 0, 0, ; Exs. 3. 0, -3, A.] 

119. Evaluation of other indeterminate forms. The evaluation 
of these forms can be made to depend on Arts. 117, 118. 

(a) The form • oo . Let f(x) and Fix) be two functions such 

that f(a) = and F(a) = oo , 

and let the limiting value of fix) • F(x) for x = a be required. 

Now f(x) • F(x) = ^ • 

This fraction has the form - when x = a, which was discussed in 

Art. 117. 

Also, fix) .F(x) = ?-&, 



which has the form — when x = a, that was discussed in Art. 118. 



EXAMPLES. 

1. Lim^Cx • cot x) = lim^ — — \ i-e. -)= lim^ — — = 1. 

tanx\ 0/ sec 2 x 

2. Determine: (1) linx^* | - — x ) tanx; (2) lini^ 



ttx r >•„„ -, ... 2" 

7T 



(3) lim^! (x — 1) tan — • \ Answers : 1, ?>i, 



(6) The form ac -oc. By combining terms and simplifying, an 
expression having the form oo — oo may be reduced to a definite 
value, or to one of the preceding illusory forms. 



188 DIFFERENTIAL CALCULUS. [Ch. XII. 

o t- ( 2 1 \ 1?w 2x-x 2 ,. 2-2x 1 

3. Lim^o = lmi x =2 — : r = nm x=^2 



x — 2 / x 2 — 4 2x 2 



4. Find : lim x=1 ( - — ) , liuto 

- 1 log x 



(1-^(1+-)}, 



lim xiao (x - vx 2 - a 2 ) . [Answers : }, f , 0.] 

(c) The forms 1*, ao° ? 0°, Suppose the function 

takes one of these forms when x = a. 

Put u = [f(x)yw. (i) 

Then log u = F(x) . log [/(*)]. (2) 

The function in the second member of (2) has one of the forms 
± • oo , oo »0, when x = a. 

Hence the limiting value of log u can be evaluated as in case (a) 
above. From this value, the limiting value of u can be derived. 
i 

5. Evaluate (1 — x)* when x — 0. (The form then is l 00 .) 

Put u = (1 — x)* i then log u = log Cj ~ ^ . 

Accordingly, lim x=M )log u = lim xd=0 ( — - — ) 

\1 -x) 

6. Find lim x=s=0 (z x )- (This form is 0°.) 
Put u = x x ; then log u = x log x. 
Accordingly, 

1 

lim I= M) log u = lim xi0 — =j- = lim z ^ _ _ 2 = lim x =o( — sc) = 
consequently, u = e° = 1 when x = 0. 

/ 1 \tanx 

7. Evaluate ( - j when x = 0. (The form then is oo°.) 

Put u=(xy anx . 



1. .-. u = - when x = 0. 



Then 



log u = tan x • log ( - } = — tan x • log x 

lim^o log u = lim xi0 ( — tan x • log x) = lim x = [ — ^-^ ) 

V cotx/ 



= lim x = 



r _i 

X 



sm 2 x 
hm xi0 



[ — cosec 2 x J x 

2 sin x cos x 



im x =o = 

lim x ± u - 1. 



119.] INDETFBMIXATE FORMS. 189 

8. Evaluate the following: (1) [1 + -J when x = oo ; (2) sin x t&nx 

i i 

when x = ; (3) x x when x — x ; (5) (1 — x) x when x = oo; (5) ( 1 + - 

V * 

/ IV — — 

whenx = cc; (6) fl+— when a; = oo ; (7) x*- 1 when x = 1 ; (8) a;*- 1 

when;c = oo; (9) x sinx when x = 0. [Jjiswrera : (1) e, (2)1, (3)1, (4)1, 
(5) x, (6) 1, (7) e, (8) 1, (9) 1.] 

9. Evaluate the folio wins: : (1) xtan x — —sec x when £ = — • 

v J 2 2 , 

(2) tana-s when re =0; (3) sec '^ = 2 tan ! when = * ; (4) sin-* s - s 
w x - sin x w 1 + cos 4 5 4 v 3 x 2 

when x = ; (5) tang when 5 = - ; (6) — - cot 2 x when x = ; 

^ ; tan 30 3' v x* 



(7) (tan^ 



when = 1; (8) (sec 0) sin( £ when = 0. [Answers 



(1) -1; (2) 2: (3) |; (4) £; (5) 3; (6) f; (7) 1; (8) 1.] 

e 

Xote. References for collateral reading on illusory forms. For a 
fuller discussion on the evaluation of expressions in these forms, and for 
many examples, see MeMahon and Snyder, Diff. Cal., Chap. V. , pages 115- 
131 ; F. G. Taylor, Calculus, Chap. XII., pages 136-148; Echols, Calculus, 
Chap. VII. ; also Gibson, Calculus, Arts. 161, 162. Eor a general treatment 
of the subject see Chrystal, Algebra, Vol. II., Chap. XXV. Eor a rigorous 
and critical treatment by the method of the calculus see Pierpont, Functions 
of Heal Variables, Vol. I., Chap. X. Also Osgood, Calculus, Chap. XI. 



CHAPTER XIII. 

SPECIAL TOPICS RELATING TO CURVES. 
ENVELOPES, ASYMPTOTES, SINGULAR POINTS, CURVE TRACING. 

Envelopes. 



120. Family of curves. Envelope of a family of curves. The 

idea of a family of curves may be introduced by an example. 
The equation 



(x — c) 2 + y 2 = 4 



a) 



is the equation of a circle of radius 2 whose centre is at (c, 0). 
If c be given particular values, say 2, 3, — 5, the equations of 
particular circles are obtained. Thus Equation (1) really repre- 
sents a family of circles, viz. the circles (see Fig. 61) whose radii 




Fig. 61. 

are 2 and whose centres are on the a>axis. The individual 
members of the family are obtained by letting c change its values 
from — oo to -f oo. A number such as c, whose different values 
serve to distinguish the individual members of a family of curves, 
is called the parameter of the family. Thus, to take another 
example, the equation y = 2 x + b represents the family of straight 
lines having the slope 2 ; and y = 2x + 5,y = 2x — 7,a.Te particu- 
lar lines of the family. (Let a figure be constructed.) In this 
case the parameter b can take all values from — oo to +cc. 

190 



120, 121.] ENVELOPES. 191 

To generalize : f(x, y, a) = (2) 

is the equation of a family of curves whose parameter is a. The 
individual members or curves of the family are obtained by giving 
particular values to a. These curves are all of the same kind, 
but differ in various ways ; for instance, in position, shape, or 
enclosed area. A family of curves may have two or more param- 
eters. Thus, y — mx -f b, in which m and b may take any values, 
has two parameters m and b, and represents all lines. The equa- 
tion (x — h) 2 + (y — k) 2 = 25, in w r hich h and k may take any 
values, represents all circles of radius 5. The equation (x — h) 2 
-f- (y — k) 2 = r, in which h, k, and r may each take any value, 
represents all circles. 

Envelope. The envelope of a family of curves is the curve, or 
consists of the set of curves, which touches every member of the 
family and which, at each point, is touched by some member of 
the family. For example, the envelope of the family of circles 
in Fig. 61 evidently consists of the two lines y — 2=0 and y+2 = 0. 
On the other hand, the family of parallel straight lines y=2x-\-b 
does not have an envelope ; and, obviously, a family of concentric 
circles cannot have an envelope. 

EXAMPLES. 

1. Say what family of curves is represented by each of the following 
equations, and in each instance make a sketch showing several members of 
the family : 

(a) x 2 + y' 2 = r 2 , parameter r. (p) y = mx + 4, parameter m. 

(c) y 2 = ±px, parameter p. (d) y 2 = ±a(x + a), parameter a. 

x 2 v 2 x 2 v 2 

(e) — J- — = 1, parameter a. (f) — 1 ^ — = 1, parameter Jc. 

w a 2 9 F w J 16 + k 92 + k ' P 

9 

(g) y = mx + — , parameter m. (h) y = mx + V25 m l + 16, parameter m. 
m 

2. Express opinions as to which of the families in Ex. 1 have envelopes, 
and as to what these envelopes may be. 

121. Locus of the ultimate intersections of the curves of a family. 
In Eq. (2), Art. 120, the equation of a family of curves, let a be 
given the particular value a Y ; then there is obtained the equation 
of a particular member of that family, viz. 

f(x,y,< h ) = 0. (1) 




192 DIFFERENTIAL CALCULUS. [Ch. XIII. 

Also, f(x, y, a x -f h) = 

is the equation of another member of the family. Let I. and II. 
be these curves. The smaller h becomes, the more nearly does 
curve II. come into coincidence with curve I. Moreover, as h be- 
comes smaller and approaches zero, A, the point of intersection of 
these curves, approaches a 

definite limiting position. For f^"^£t-2- 

example, if (Fig. 61) the centre 
L approaches nearer to C, then 
K, the point of intersection of 
the circles whose centres are 

at C and L, moves nearer to jS yig. 62. 

P; and finally, when L reaches 

C, K arrives at the definite position P. The locus of the limiting 
position of the point (or points) of intersection of two curves of 
a family which are approaching coincidence is called the locus of 
ultimate intersections of the curves of the family. For instance, in 
the case of the family of circles in Fig. 61, this locus evidently 
consists of the lines y — 2 = and y + 2 = 0. 

Note. The last-mentioned locus may also be derived analytically. 

Let ( x _ Cl ) 2 + 2/ 2 = 4 (1) 

and (x - d - h)' 2 + y 2 = 4 (2) 

be two of the circles. On solving these equations simultaneously in order to 
find the point of intersection, there is obtained 

(x — ci) 2 — (x — ci — h)- = ; whence ^(2 x — 2 c\ — h) = 0, 

and, accordingly, x = C\ -\ — 

A 

An ultimate point of intersection is obtained by letting h approach zero. 
If h = 0, then x = c l5 and by (1) y = ± 2. Thus y = ± 2 at the ultimate 
points of intersection, and therefore the locus of these points is the pair of 
lines y = ± 2. 

N.B. In the following articles "the locus of ultimate intersections" is 
denoted by I. u. i. 



121, 122.] 



ENVELOPES. 



193 



122. Theorem. In general, the locus of the ultimate intersections 
touches each member of the family. Let L, II., III. be any three 
members of the family, and let I. and II. intersect at P, and II. 
and III. at Q. When the curve I. approaches coincidence with 
II., the point P approaches a definite position on I. u. i. of the 
curves of the family. When the curve III. approaches coincidence 
with II., Q approaches a definite position on I u. i. When I. and 
III. both approach coincidence with II., P and Q approach each 
other along II., and at the same time approach I. u. i. When P 




and Q finally reach each other on II., they are also on I. u. i. More- 
over, when P and Q come together, the tangent to II. at P and the 
tangent to II, at Q come into coincidence as a line which is at the 
same time a tangent to curve II. and a tangent to I. u. i. at the point 
where P and Q meet. Thus the curve II. and I. u. i. have a com- 
mon tangent at their common point. Similarly it can be shown 
that I. u. i. touches every other curve of the family. Since, in gen- 
eral, each point of I. u. i. may be approached in the manner indicated 
in this article, the above theorem may be thus supplemented: In 
general, /. u. i. is touched at each of its points by some member of 
the family. 

Note 1. The family of circles, Fig. 61, will serve to illustrate this theorem. 

Note 2. An analytical proof of the theorem is given in Art. 123, Note 3. 

Note 3. It is necessary to use the qualifying phrase in general in the 
enunciation of the theorem, for there are some families of curves (viz. curves 
having double points and cusps, see Arts. 129, 130), in which a part of I. u. i. 
may not touch any member of the family. It is beyond the scope of this 
book to go into these cases in detail. (See Edwards, Treatise on the Biff. Cal., 
Art. 365 ; Murray, Differential Equations, Chap. IV.) Illustrations may be 
obtained by sketching some curves of the families (y + c)' 2 = x 3 and 
{y + c) 2 = x(x - 3) 2 . 



194 DIFFERENTIAL CALCULUS. [Ch. XIII. 

123. To find the envelope of a family of curves having one pa- 
rameter. It is in accordance with the definitions and theorem 
in Arts. 120-122 to say that the envelope of a family of curves 
f{x, y, a) = 0, if there be an envelope, is, in general, the locus of the 
limiting position of the intersection of any one of the curves of the 

family, say the curve 

f(x,y,a) = (1) 

with another curve of the family, viz. 

f(x,y,a + Aa) = (2) 

when the second curve approaches coincidence with the first; that 
is, when Aa approaches zero. 

From (1) and (2), f(x, y,a + Aa)-f(x, y,a) = 0; 

hence f(x,y,a + Aa)-f(x,y, a) = ^ 

Aa v J 

Now Equations (1) and (3) may be used, instead of (1) and (2), 
to find the points of intersection of curves (1) and (2). If Aa = 0, 
the point of intersection approaches an ultimate point of inter- 
section. When (Arts. 22, 79) Aa = 0, Equation (3) becomes 

fafi x >y> a ) =°- ( 4 ) 

Thus the coordinates x and y of the point of ultimate inter- 
section of curves (1) and (2) satisfy Equations (1) and (4) ; and, 
accordingly, satisfy the relation which is deduced from (1) and 
(4) by the elimination of a. Hence, in order to find the equation 
of I. u. i. of the family of curves f(x, y,a) = eliminate a between 
the equations 

f(x, y,a)=0 and ^ f(x, y, a) = 0. (5) 

The result obtained is, in general, also the equation of the 
envelope. 

Note 1. A slightly different way of making the above deduction is as 
follows. Let the equations of two curves of the family be 

fix, y,d) = (6), and f(x, y,a + h) = 0. (7) 



123.] ENVELOPES. 105 

By Art. 108, Eq. (3), Equation (7) may be written 

/(x, y, a) + h-^ /(x, y, a + Oh) = 0, in which | 6 1< 1. (8) 

By virtue of (6) this becomes ~-/(x, ?/, a + 0ft) = 0. (9) 

Accordingly, the coordinates of the intersection of curves (6) and (7) 
satisfy (6) and (9). When h becomes zero, the point of intersection becomes 

an ultimate point of intersection. Hence the ultimate points of intersection 

f\ 
satisfy equations /(x, y, a) = and — f(x, y, a) = 0, and, accordingly, the 
a-eliminant of these equations.* 

Note 2. For an interesting and useful derivation of result (5) for cases 
in which /(x, ?/, a) is a rational integral function of a, see Lamb's Calculus, 
Art. 157. 

Note 3. To show that, in general, the a-eliminant of Equations (5) touches 
any curve of the family. 

Let the second of Equations (5) on being solved for a give a = 0(x, y). 
Then the equation of the I. u. i. of the family of curves /(x, y, a) = is 

f(x, y, a) = in which a = <f>(x, y). (10) 

The slope -^ of any one of the family of curves /(«, y, a) = is given (see 
Art. 56), by the equation qj- qs ^/ 

dx dy dx~ ^ ' 

The slope — of the I. u. i. is obtained from Equations (10). On taking 
the total x-derivative in the first of these equations, 

dx dy dx Ba dx ' * J 

But by the second of (5), ~ = 0, and accordingly, (12) reduces to 

% + f a i = *. 03) 

dx dy dx K ' 

Thus the slope of the I. u. i. and the slope of any member of the family 
are both given by the same equation. Hence, at a point common to any 
curve and the I. u. t, the slopes of both are the same, and accordingly, the 
curve and the I. u. i. touch at that point. 

Sometimes the value of -^ obtained from (11) is indeterminate in form, 

dx K J 

and the slopes of the curve and I. u. i. may not be the same. See Arts. 131, 
122 (Note 3), and Lamb, Calculus, Art. 158. 

* This method of finding envelopes appears to be due to Leibnitz. 



196 DIFFERENTIAL CALCULUS. [Ch. XIII. 

EXAMPLES. 

1. Find the envelope of the family of circles (see Art. 120) 

(x - c) 2 + y* = 4. (1) 

Here, on differentiation with respect to the parameter c, 

2 (x - c) = 0. (2) 

The elimination of c between these equations gives 

*/ 2 = 4, 
which represents the two straight lines y = 2, y = — 2. 

2. Find the envelope of the family of lines 

y = mx — 2 pm — pm 3 , (1) 

in which m is the parameter. (This is the equation of the general normal of 
the parabola y' 2 = 4 px ; see works on analytic geometry.) On differentiation 
with respect to the parameter to, 

= x - 2p - 3pm?. (2) 

The TO-eliminant of (1) and (2) is the equation of the envelope. 
On taking the value of to in (2) and substituting it in (1), and simplifying 
and removing the radicals, there is obtained 

27 py 1 = 4 (as- 2 p)K (3) 

Note 4. In Art. 104 it is shown that the normals to a curve touch its 
evolute. It also appears from Art. 104 that each tangent to an evolute is 
normal to the original curve. Accordingly, it may be said that the evolute 
of a curve is the envelope of its normals, and likewise that the evolute of a 
curve is the I. u.i. of its {family of) normals. (See Art. 104, Note 2, and 
Art. 101, Note 5.) 

Note 5. Compare Ex. 1, Art. 103, Ex. 2 above, and Ex. 1, Art. 124. 

3. If A, B, C are functions of the coordinates of a point and m a 
variable parameter, show that the envelope of Am™ + Bm + C = is 
B2-4AC = 0. 

Note 6. The result in Ex. 3 is the same in form as the condition that the 
roots of the quadratic equation in m be equal. This result is immediately . 
applicable in many instances. It is very easily deduced on taking the point 
of view explained in the article mentioned in Note 2. 

4. Deduce the result in Ex. 3 without reference to the calculus. 
Apply this result to Ex. 1. 



123, 124.] ENVELOPES. 197 

N.B. Make figures for the following examples. 

5. Find the curves whose tangents have the following general equations, 
in which to is the variable parameter : 

(1) y = mx + a Vl + to 2 . (2) y = mx + Va'h.i 2 + b 2 . 

(3) y = mx± Vam 2 + bm + c. (4) y = mx + aVm. 



(5) m*x = my + a. (6) y — b = m(x - a) + r Vl + to' 2 . 

6. Find the envelopes of the following lines : 

(1) x sin — y cos 6 + a = 0, parameter 0. (2) as + y sin = a cos 0, 

parameter 0. (3) ax sec a — by cosec oj = a 2 — 6 2 , parameter ct. 

7. Find the envelopes of (1) the parabolas !/ 2 = 4a(x-«), parameter a ; 
(2) the parabolas cy 2 = a 2 (x — a), parameter a. 

8. Show that if A, B, C are functions of the coordinates of a point, and 
a a variable parameter, the envelope of A cos a + B sin a = C is A 2 + B 2 = G' 2 . 

9. Find the evolute of the ellipse x = a cos <p, y — b sin 0, considering 
the evolute of a curve as the envelope of its normals. 

10. One of the lines about a right angle passes through a fixed point, and 
the vertex of the angle moves along a fixed straight line : find the envelope 
of the other line. 

11. From a fixed point on the circumference of a circle, chords are 
drawn, and on these as diameters circles are described. Show that they 
envelop a cardioid. 

124. To find the envelope of a family of curves having two parame- 
ters. Let r/ ,. A 
f(x, y, a, b) = 

be a family of curves which has two parameters. If there is a 
given relation between these parameters, say 

F(a, b) = 0, 

then the two parameters practically come to one, and accordingly, 
the case reduces to that considered in Art. 123. 



EXAMPLES. 

1. Find the envelope of the normals to the parabola y 2 = ipx. The 
equation of the normal at any point (x±, y{) on this parabola is 

y-y 1 +^(x~x 1 ) = 0. 
dpi 



198 DIFFERENTIAL CALCULUS. [Ch. XIII. 

This reduces to 2py — 2py,_ + xyi — Xiyi = 0. (1) 

Here there are two parameters, x\ and y\. They are connected by the 

relation „ , 

y ± 2 = ±px x . 

Hence (1) becomes 2 py — 2 pyi + xy\ — ^- = 0, (2) 

which involves only a single parameter y\. On differentiating in (2) with 
respect to the parameter y x and then eliminating y h there will appear the 
equation of the envelope, viz. 

27 py 2 = 4 (x- 2 p)K 
Compare Ex. 1 with Ex. 1, Art. 103, and Ex. 2, Art. 123. 

Note. This problem may be expressed : Find the envelope of the line 
(1), given that the point (a?i, y{) moves along the parabola y 2 = kpx. 

2. Find the envelope of the line 

- + f = l (1) 

a b 

when the sum of its intercepts on the axes is always equal to a constant c. 

Since a + b = c, (2) 

Equation (1) may be written - H ^— = 1, 

a c — a 

i.e. (c — a)x + ay = ac — a 2 . (3) 

Thus (1) is transformed into an equation involving a single parameter a. 
On differentiating in (3) with respect to the parameter a, 

— x + y = c — 2 a. (4) 

The elimination of a between (3) and (4) gives 

x 2 + y 2 + c 2 = 2 ex + 2 xy + 2 cy. 

This reduces to Vx + Vy = Vc. 

See Ex. 7, Art. 62. 

The elimination of a and b can also be performed thus : 

Differentiation in (1) and (2) with respect to a gives 

_x_il #_oandl+- = 0. 



On equating the values of 

da 



b 2 da da 

db 



x 



Vy 



= M- ; whence - = -^- (5) 

i 2 b 2 ' a Vx 



124, 125.] ASYMPTOTES. 199 

From (2) and (5), a = cV ^ , b = _ ^5 — 

Vx + Vy Vx + Vy 

On substitution in (1) and reduction, Vx + Vy = Vc. 

This second method is generally more useful than that used in Ex. 1 and 
in the first way of working Ex. 2, in cases when the two parameters are 
involved symmetrically in the equation and in the expression of the relation 
between the parameters. 

3. Eind the envelope of the straight lines the product of whose intercepts 
on the axes of coordinates is equal to a 2 . 

4. Eind the envelope of a straight line of fixed length a which moves with 
its extremities in two lines at right angles to each other. 

5. A set of ellipses which have a common centre and axes, and in 
which the sum of the semi-axes is equal to a constant a, is drawn : find the 
envelope of the ellipses. 

6. Show that the envelope of a family of co-axial ellipses having the 
same area consists of two conjugate rectangular hyperbolas. 

7. Circles are described on the double ordinates of the parabola 
y 2 = 4 ax as diameters : show that the envelope is the equal parabola 
y 2 = 4a(x + a). 

8. Circles are described having for diameters the double ordinates of 
the ellipse whose semi-axes are a and b : show that their envelope is the 
co-axial ellipse whose semi-axes are Va 2 + b 2 and 6. 

9. About the points on a fixed ellipse as centre, ellipses are described 
having axes equal and parallel to the axes of the fixed ellipse : show that 
their envelope is an ellipse whose axes are double those of the fixed ellipse. 

10. A straight line moves so that the sum of the squares of the perpen- 
diculars on it from two fixed points (± c, 0) is constant (= 2 k 2 ) : show that 

x 2 v 2 

its envelope is the conic f- — = 1- 

* k 2 - c 2 k 2 

11. If the difference of the squares in Ex. 10 is constant, show that the 
envelope is a parabola. 

12. Show that if the corner of a rectangular piece of paper be folded 
down so that the sum of the edges left unfolded is constant, the crease will 
envelop a parabola. 

Asymptotes. 

125. Rectilinear asymptotes. In preceding studies acquaint- 
ance has been made with two lines related to the hyperbola, 
called asymptotes and possessing the following properties : 
(a) These lines are the limiting positions which the tangents to 
the hyperbola approach when the points of contact recede for an 



200 DIFFERENTIAL CALCULUS. [Ch. XIII. 

infinite distance along the curve (or, as it may be expressed, 
recede towards infinity) ; (6) the lines themselves do not lie 
altogether at infinity. (This is the mathematical way of saying 
that the lines run across the field of view ; in fact, in the case of 
the hyperbola they pass through the centre of the curve.) 

Besides hyperbolas there are many other curves which have 
branches extending to an infinite distance and which have associ- 
ated with them certain lines having properties like (a) and (6) ; 
namely, lines : (1) that are the limiting positions which the tan- 
gents to the infinite branches approach when the points of contact 
recede towards infinity ; (2) that do not lie altogether at infinity ; 
for instance, using rectangular coordinates, lines that pass within 
a finite distance of the origin. 

Lines having properties (1) and (2) are called asymptotes of the 
curves. Thus an ellipse cannot have an asymptote, since it has 
no branch extending to infinity (see Ex. 3, Art. 127). Again 
the parabola y 2 = 4px has no asymptote, for (see Ex. 4, Art. 127) 
the tangent at an infinitely distant point of this parabola crosses 
each of the axes of coordinates at an infinite distance from the 
origin, and, accordingly, no part of this tangent can be in sight ; 
i.e. it lies wholly at infinity. (The asymptotes are apparent in 
the figures on pages 460-464.) 

It will now be shown how an examination may be made for the 
asymptotes of curves whose equations have the form 

F(x,y)=0, (1) 

where F(x, y) is a rational integral function of x and y. For this 
it is necessary to call to mind the algebraic property stated in the 
following note. 

Algebraic Note. On substituting - for x in the rational integral equation 

c x n + dx n - 1 + c 2 x n - 2 + ••• + c n -\x + c n = 0, (a) 

and clearing of fractions, it becomes 

Co + ctt + c- 2 t 2 + .» + Cn-it"- 1 + c n t n = 0. (6) 

It is shown in algebra that if a root of Equation (b) approaches zero, cq 
approaches zero ; and that if a second root also approaches zero, c x also 

approaches zero. But, since x — -, when a root of (6) approaches zero, a 



125. 126.] ASYMPTOTES. 201 

root of (a) increases beyond all bounds, i.e., to use a common phrase, it 
approaches infinity. Hence, the condition that a root of (a) approach 
infinity is that c approach zero, and the condition that a second root of (a) 
at the same time approach infinity is that c\ also approach zero ; and so on 
for other roots approaching infinity. This is briefly expressed by saying that 
equation (a) has a root equal to infinity when c = 0, and has two roots 
equal to infinity when c Q = and Ci = 0. 

126. To find asymptotes which are parallel to the axes of coordi- 
nates. Suppose that the equation of the curve F(x, y) = [Art. 
125 (1)] is of the nth degree, and that the terms in the first mem- 
ber of this equation are arranged according to decreasing powers 
of y. Then the equation has the form 

PoV n + W n_1 + P-2U n ~ 2 + • • • 4- Pn-lV +p n = 0. (1) 

Here, p is a constant ; p 1 may be an expression in x of the first 
degree at most, say ax -f- b ; p 2 may be of the second degree at 
most, say ex 2 + dx -f- e ; p z may be of the third degree in x at 
most ; • • • ; and p n may be of the nth degree in x at most. For 
if any one of the respective p's were of a higher degree than that 
specified above, F(x, y) would be of a higher degree than the nth. 

Ex. 1. Arrange the first members of the following equations (a) in 
descending powers of x ; (6) in descending powers of y : 

(1) xy - ay - bx = 0. (2) x 3 + xy 2 + 2 x 2 - 2 y 2 - 7 x + 4 y - 11 = 0. 

(3) 2 xy* - x^y + 3 tf- - 3 x 2 + 4 xy - 2 x + 7 y + 1 = 0. 

(4) yz + x*y + x 2 + 2 xy + 7 x + 2 = 0. 

Xow suppose that in (1) j9 = ; then (1) may be written 
. f -f (aa; + 6) 2 /"- 1 + (co: 2 + cfc + e)?/"- 2 +psir-* +»• 

+ Pn-iy+Pn = 0. (2) 

If this be regarded as an equation of the nth degree in y, then 
to any finite value of x there correspond n values of y, one of 

which is infinitely great. If also ax -(-6 = 0, i.e. if x — , a 

second of the n values of y is infinitely great. In a similar way 
points whose abscissas are infinitely great and whose ordinates are 
finite may be found. 



202 DIFFERENTIAL CALCULUS. [Ch. XIII. 

Ex. 2. Thus in Ex. 1 (1) the equation, which is of the second degree, may- 
be written y(x — a) — bx = 0. Accordingly one value of y is infinite ; a second 
value of y is infinite when x = a. 

Ex. 3. Show that a second value of x is infinite when y = b. 

It will now be shown that an infinite ordinate whose distance 
from the origin is finite is tangent to the curve at the infinitely dis- 
tant point. 



On differentiating in (2) with respect to x and solving for -^, 

dx 

dy _ ay 71 - 1 4- (2 ex + d)y n ~ 2 +•••+#'» 

dx ~ ~ O - 1) (ax + b)y n ~ 2 + (n — 2)(cx 2 + dx + e)y n ~ s H \- p n - 



(3) 



When x = — , the numerator in the second member is an infinity of an 

a 
order at least two higher than the denominator, and hence the value of the 

fraction is then infinite. Hence the line x = is a tangent at any point 

b a 

for which x = and y = oo. 

a 
In a similar way it can be shown that if one of the values of x in Equa- 
tion (1), Art. 125, is infinite when y = c, in which c is finite, then y = c is 
a tangent at any point for which x = oo and y = c. 

Note 1. If [see Eq. (2)] x = — also satisfies ex 2 + dx + e = 0, then 

a 
three values of y in F (x, y) = are infinitely great for this value of x. The 

line x — is then an inflexional tangent (see Art. 78, Note 1) at infinity. 

a 
Note 2. This method of finding asymptotes parallel to the axes can be 
applied to curves whose equations are not of the kind considered above. 
Instances are given in Exs. 7, 8 (6), (9) that follow. 

EXAMPLES. 

4. Eind the asymptotes of the curves in Ex.. 1. 

5. Determine the finite points (if they exist) in which each asymptote 
in Ex. 4 meets the curve to which it belongs. 

6. Show that the line x — a is an asymptote of the curve y = -ri^l 
when 0(a) and </>'(a) are finite. 

Here, \im x ± a y = oo. Also ^ = (x- a)4>>(x)- ct>(x) . whence lim ^ = ^ 

dx (x — a) 2 dx 

Hence x = a is a tangent at an infinitely distant point (x = a, y = oo). 

7. Examine y = tan x for asymptotes. 

Here y = + oo when x = -, — , — , .... 

y 2' 2 2 

Also, ^ = sec 2 x. Hence ^ = oo when x = -, ^ ^, .... 
dx dx 2 2 2 

.•. x = -, a; = — , # = — -, •••, are asymptotes. 



126, 127.] ASYMPTOTES. 203 

8. Determine the asymptotes of the following curves : (1) The hyper- 
bola xy = a 2 . (2) The cissoid y 2 = — — — (3) The witch y 



2a — x x 2 + 4 a 2 

(4) (x 2 - a 2 ) (y 2 - b 2 ) = a 2 b 2 . (5) a 2 x = y(x - a) 2 . (6) y = log x. (7) y = e*. 
(8) The probability curve y = e~ x2 . (9) y = sec x. 

127. Oblique asymptotes. There are asymptotes which are not 
parallel to either axis. The method of finding them can best be 
shown by an example. 

EXAMPLES. 

1. Find the asymptotes of the folium of Descartes (see page 463) 

x s + y 3 = 3 a xy. (1) 

First find the intersections of this curve and the line 

y = mx+b. (2) 

On solving these equations simultaneously, 

(1 + m 3 )£ 3 + 3 (m 2 b - am)x 2 + 3 (mb 2 - ab)x + 6 3 = 0. 

Line (2) is a tangent to the curve (1) at an infinitely distant point, if two 
roots of this equation are infinitely great. That is, if 

1 + m 3 = 0, and m 2 b - am = 0. (3) 

That is, on solving Equations (3) for m and 6, if 

m = — 1, and b = — a. 

Hence, the asymptote is y + x + a = 0. 

Note 1. A curve whose equation is of the nth degree has n asymptotes, 
real or imaginary. This may be apparent from the preceding discussion. 
For proof of this theorem see references for collateral reading, Art. 128. 

In Ex. 1 two values of m in Equations (3) are imaginary ; thus curve (1) 
has one real and two imaginary asymptotes. 

2. Find the asymptotes of the hyperbola 5 2 x 2 — a 2 y 2 = a 2 b 2 . 

3. Show by the method used in Ex. 1 that the ellipse b 2 x 2 + a 2 y 2 = a 2 b 2 
has no real asymptotes. 

4. Show by the method used in Ex. 1 that the parabola y 2 = 4px 
does not have an asymptote. 



204 DIFFERENTIAL CALCULUS. [Ch. XIII. 

5. Find the asymptotes of the following curves: (1) y 3 = x 3 + x. 
(2) x 4 - yi - 3 X s - xy 2 - 2 x + 1 = 0. (3) xy(y - x) = 3 x 2 + 2 y* 
(4) (x 2 -?/2)2 _ 4 y 2 + y .+ 2 a- + 3 = o. (5) x 3 - 8 y* + 3 x 2 - xy - 2 y 2 = 0. 

Note 2. Other methods of finding asymptotes. 

a. Find the values of the intercepts on the axes of coordinates of the 
tangent at a point (x', y') on a curve [see Art. 61, Equation (3)], when 
x' — oo, or y' = co, or both x' and y' are infinitely great. If one or both of 
these intercepts is finite, the tangent is an asymptote. Its equation can be 
written on finding its intercepts. 

6. Apply this method to Exs. 2, 4, above. [See Note, p. 212.] 

b. Find the length of the perpendicular from the origin to the tangent 
at (x', y') when x' = co, or y' = oo, or both x' and y f are infinitely great. 
If this length is finite, the tangent is an asymptote. 

7. Do Exs. 2, 4, by this method. [See Note, p. 212.J 

c. By means of the equation of the curve express y in terms of a series 
in decreasing powers of x, or express x in terms of a series in decreasing 
powers of y. From one of these expressions there may sometimes be de- 
duced the equation of a straight line which, for infinitely distant points, 
closely approximates to the equation of the curve. 



8. Thus, in the hyperbola in Ex. 2, 

* « 2 a \ x 2 ) 



a \ 2 x 2 / a x 4 x 3 

It is apparent from this that the farther away the points on the lines 

y = ± — are taken, the more nearly will they satisfy the equation of the 
a 

hyperbola, and that when x increases beyond all bounds, the points on these 

lines satisfy the equation of the hyperbola. Accordingly, these lines are 

asymptotes. 

Note 3. Curvilinear asymptotes. Expansion may sometimes reveal 
the equation of a curve of higher degree than the first whose infinitely distant 
points also satisfy the equation of the given curve. Accordingly the two 
curves coincide at infinitely distant points. The two curves are said to be 
asymptotic, and the new curve is called a curvilinear asymptote of the 
original curve. For a discussion on curvilinear asymptotes see Frost's Curve 
Tracing, Chaps. VII. and VIII. 



127, 128.] 



ASTMPTOTES. 



205 



128. Rectilinear asymptotes : polar coordinates. In order to find 
the asymptotes of the curve 



/(,-, 0)=O 



(1) 



a method similar to that outlined in Art. 127, Note 2 (6), can be 

used. First find the value of 6 
in Equation (1) for when the 
radius vector r is infinitely great. 
Suppose that this value of is 
6 X . Thus the point (r=oo, 0=0 X ) 
is an infinitely distant point of 
the curve. If the tangent TN at 
this infinitely distant point is 
an asymptote, it passes within 
a finite distance from 0. Accord- 
ingly, TN is parallel to the radius 




Fig. 64. 



vector, and the subtangent OM, viz. 

EXAMPLES 



i a — (Art. 64) is finite for 



1. Find and draw the asymptote to the reciprocal spiral rd = a. 



Here 



.-. r = co when 8 = 0. 



Also 



dd 
dr 



cW 
dr 



i. (See Fig., page 
464.) 

Hence the asymptote is parallel to the initial line and at a distance a to 
the left of one who is looking along the initial line in the positive direction. 

Note 1. The convention used in Ex. 1 is as follows : A positive subtan- 
gent is measured to the right of a person who may be looking along the 
infinite radius vector in its positive direction, and a negative subtangent is 
measured toward the left. 

2. Find and draw the asymptotes to the following curves : (1) r sin d 



= ad. (2) r cos 6 = a cos 2 0. (3) 



r sin -= a. 

2 



Xote 2. Circular asymptotes. If the radius vector r approaches a fixed 
limit, a say, when 6 increases beyond all bounds, then as 6 increases, the curve 
approaches nearer to coincidence with the circle whose centre is at the pole 
and whose radius is a. This circle, whose equation is r = a, is said to be a 
circular asymptote, or the asymptotic circle, of the curve. 



206 DIFFERENTIAL CALCULUS. [Ch. XIII. 

3. In the reciprocal spiral, Ex. 1, if d = co, then r = 0. Hence the 
asymptotic circle is a circle of zero radius, viz. the pole. 

a 

4. Find the rectilinear and the circular asymptote of r = 

References for collateral reading on asymptotes. McMahon and 
Snyder's Diff. Col., Chap. XIV., pages 221-242 ; F. G. Taylor's Calculus, 
Chap. XVI., pages 228-249, and Edwards's Treatise on the Differential Cal- 
culus, Chap. VIII., pages 182-210, contain interesting discussions on asymp- 
totes, with many illustrative examples. For a more extended account of 
asymptotes see Frost's Curve Tracing, Chaps. VI. -VIII., pages 76-129. 



Singular Points. 

129. Singular points. On some curves there are particular 
points at which the curves have certain peculiar properties which 
they do not possess at their points in general. For instance, there 
are points of maximum or minimum ordinates (Art. 75), points of 
inflexion (Art. 78), and points of undulation (Art. 78). There are 
also points through which a curve passes twice or more than twice 
(see Figs. 65 a, b, c), and at which it has two or more different 
tangents ; there are points through which pass two branches of a 
curve that have a common tangent (Figs. 66 a, b, c, d) ; and there 
are other peculiar points hereafter described. Points of maximum 
and minimum ordinates depend on the relative position of a curve 
and the axes of coordinates ; the peculiarities at the other points 
referred to above are independent of the axes and belong to the 
curve whatever be its situation. Points at which a curve has 
peculiarities of this kind are called singular points. Some of these 
singular points are considered in Arts. 130, 131 . 

130. Multiple points. Double points. Cusps. Isolated points. 
Multiple points are those through which a point moving along the 
curve, while changing the direction of its motion continuously, 
can pass two or more times, and at which the curve may have two 
or more different tangents. 

For example, in moving from L to M along the curves in Figs. 
65 a, b, c, a point passes through A and C three times and through 
B and D twice. At A there are three different tangents, at C 
there are three, and at B and D there are two each. Points, such 



128, 130.] 



SINGULAR POINTS. 



207 



as B and D, through which the point moving along the curve, 
while continuously changing the direction of its motion, can pass 





2 / 1 

M 



R 



Fig. 6" a. 



Fig. 65 b. 



Fig. 65 c. 



twice, are called double points; points such as A and C are called 
triple points. The curve r = a sin 2 (see p. 464) has a quadruple 
point. 

Note 1. Multiple points are also called nodes. (Latin nodas, a knot.) 

Cusps are points where two branches of the curve have the same 
tangent. See Figs. 66 a, b, c, d. 

In Fig. 66 a both branches of the curve stop at A and lie on 
opposite sides of their common tangent at A. In Fig. 66 b both 
branches stop at B and lie on the same side of the tangent at B. 
Both branches of the curve pass through C. Accordingly C is 
sometimes called a double cusp. If a point is moving along a 
curve LKM which has a single cusp at K (Fig. 66 d), there is an 




Fig 66 a. 



Fig. 66 b. 



Fig. 66 c. 



Fig. 66 d. 



abrupt (or discontinuous) change made in the direction of its 
motion on its passing through K. On arriving at K from L the 
moving point is going in the direction a; on leaving iTfor ilfthe 
moving point is going in the direction b. Thus at K it has sud- 
denly changed the direction of its motion by the angle it. 

Note 2. A cusp such as K (Fig. QQ d) may be supposed to be the final 
(or limiting) condition of a double point like D (Fig. 65 c) when the loop 
BR dwindles to zero and the two tangents at D become coincident. 



208 DIFFERENTIAL CALCULUS. [Ch. XIII. 

Isolated or conjugate points are individual points which satisfy 
the equation of the curve but which are isolated from (i.e. at a 
finite distance from) all other points satisfying the equation. 

EXAMPLES. 

1. Sketch the curve y 2 = (x - d)(x - b)(x - c), in which a, 6, and c, 
are positive and a<b<.c. 

2. Sketch the curve y 2 = {x - a)(x - b) 2 , in which a<b and both 
are positive. 

3. Sketch the curve y 2 = (x — a) 2 (x — 6), in which a and b are as in Ex. 2. 

4. Sketch the curve y 2 = (x — a) 3 , in which a is positive. 

The sketch in Ex. 1 will show an oval from x = a to x = 6, a blank space 
from x=b to x=c, and a curve extending from x = c to the right. The sketch 
in Ex. 2 will show a curve having a double point at (6, 0). The sketch in 
Ex. 3 will show a conjugate point at (a, 0), a blank space from x=a to x=b, 
and a curve extending from x= b to the right. The sketch in Ex. 4 will show 
a curve having a cusp at (a, 0). 

Note 3. Other singular points. There also are points called salient 
points, like D (Fig. 98), for instance, where two branches of the curve stop 
but do not have a common tangent. In these 
cases the slope of the tangent changes abruptly. 
Accordingly, if y — 0(x) be the equation of the 
curve, 4>' (x) is discontinuous at the salient 
points. (See Exs. 5, 6, below.) A salient point 

such as D may be considered to be the limiting condition of a double 
point like D (Fig. 96 c), when the loop DB dwindles to zero but the two 
tangents at D do not become coincident. (Compare 
"A Note 2.) 

There are also stop points, as A, Fig. 68, where the 
j?i^. do. curve stops and has but one branch. See Ex. 7. 

i 

5. In the curve y(l + e x ) = x show that when x approaches the origin 
from the positive side, the slope is zero ; if from the negative side, the slope 
is 1. The origin is thus a salient point. Suggestion : The slope at the 

origin may be taken as lim^ -• 1 Find the angle between the branches at the 
origin. x J 

. 6. In the curve y = x e ~ show that when x approaches the origin 

£+.1 
from the positive side the slope is 4- 1, and if from the negative side, the 
slope is — 1. The origin thus is a salient point : find the angle between the 
branches there. 

7. Show that the origin is a stop point in the curve y = x log x. 





130, 131.] SINGULAR POINTS. 209 

131. To find multiple points, cusps, and isolated points. From 
Art. 130 it is evident that in order to determine the character of 
a point on a curve, it is first of all necessary to examine the tan- 
gent (or tangents) there. Let the equation of the curve be 

A»nr) = 0, (l) 

and let /(a, y) be a rational integral function of x and y. Then 

df 

| = -|. [Art. 84, (4).] (2) 

By 

Now at a multiple point or a cusp -M- has not a single definite 

dx 

value, and, accordingly, at such points — in (2) must have an 

/-> cix 

indefinite form, viz. the form -•* Hence, at a multiple point of 

curve (1) 

^=0 and 3/=0. (3) 

dx By . } 

The solutions of Equations (3) will indicate the points which it 
is necessary to examine, f At these points 



dx 0' 



W 



the indefinite form in the second member can be evaluated by the 
method explained in Chapter XII., Art. 117, and applied in Note 
below. $ Suppose that the second member of (4) has been evaluated 

and the resulting equation solved for -^- Then : If — has two 

dx dx 

real and different values at the point under consideration, the 
point is a double point or a salient point; if — has three real 

and different values there, it is a triple point ; and so on. If — 

dx 

* This is frequently called an "indeterminate" form. The evaluation of 
(so-called) " indeterminate forms" is discussed in Chapter XII. 

t The values of x and y that satisfy Equations (3) may give points that 
are not on the curve. Of course these points need not be examined further. 

t Or by other methods referred to in Art. 114. 



210 



DIFFERENTIAL CALCULUS. 



[Ch. XIII. 



has two real and equal values at the point which is being examined, 

the point is a cusp. If -^ has imaginary values at the point, it 
is an isolated point. J 

If the point is a cusp, the kind of cusp can be found by further examina- 
tion of the curve in the neighborhood of 
the point. For example, if (xi, y{) is 
known to be a cusp and it is found that 
for x = Xi — h (h being infinitesimal), y is 
imaginary, then the curve does not extend 
through (xi, ?/i ) to the left, and thus the 
cusp is not a double cusp. If for x = xi + h, 
the value of the ordinate of the tangent at 
(%u Vi) is less than the ordinates of both 
branches of the curve, the cusp is as in Fig. 
69. In a similar way tests may be devised 
and applied in special cases as they arise. 




Fig. 69. 



Note. The evaluation of the second member of Equation (2) gives, 
by Art. 117, and Art. 81, (5) 



dx* 



d 2 f a y 

)y d% dx 



dx 



d 2 f + d 2 fdy 



(5) 



dxdy dy 2 dx 



If the second member of (5) is not indefinite in form, this equation, on 
clearing of fractions and combining, becomes 



d 2 f(dyy d 2 f dy d 2 /_ ft 

dy 2 \dx) "*" dydxdx ^ dx 2 ~ ' 



(6) 



a quadratic equation in 
dy 



dy 

dx' 



By the theory of quadratic equations, the two 



values of ~ are real and different, real and equal, or imaginary, according as 

/ Q2f \2 ^x - 2 , q 2 ~ 

\ ^~ir ) is respectively greater than, equal to, or less than -— • ^. Hence, 
the point is a double point, a cusp, or a conjugate point, according as 



\dydx ^' 



or < 



dy 2 ' dx 2 ' 



If the second member of (5) also is indefinite in form, proceed as required 

by Art. 117, remembering that =&■ here is constant. The resulting equation 

dx 



will be of the third degree in 



dy 

dx 



131, 132.] SINGULAR POINTS. 211 



EXAMPLES. 

1. Examine the curve x 3 — y 2 — 7 x 2 + 4 y + 15 x — 13 = for singular 
points. 

Here # = _3x 2 - 14x + 15. (1) 

dx -2 ?/ + 4 w 

On giving each member the indefinite form -, and solving the equations 

3x 2 -14x + 15 = 0, 
-2y + 4 = 0, 

it results that x = 3 or f, and y = 2. 

Substitution in the equation of the curve shows that x = f , y = 2, do not 
satisfy the equation, and that x = 3, ?/ = 2 do. Accordingly, the point (3, 2) 
is the point to be further examined. 

On evaluating, by the method shown in Chap. XII., the second member 
of (1) for the values x = 3, y = 2, it is found that 

dy 6x-14 . (dy\ 2 . dy ^ 

£=-^w' whence {£) =' 2 ' and £= ±vs - 

dx 

Thus the curve has a double point at (3, 2), and the slopes of the tangent 
there are + V2 and — V2. 

[The curve consists of an oval between the points (1, 2), and (3, 2), and 
two branches extending to infinity to the right of (3, 2).] 

2. Sketch the curve in Ex. 1. 

3. Examine the following curves for singular points : 

(1) a 2 y 2 = x\a 2 - x 2 ). (2) x 3 + 9 x 2 - y 2 + 27 x + 2 y + 26 = 0. 

(3) y s - x 2 - 3 y 2 + 3 y + 4 x - 5 = 0. (4) The curve in Ex. 5 (5), Art. 127. 

(5) x 3 + ?/ 3 + 3 x°-y + 3 xy 2 - 10 y 2 - 16 xy - 10 x 2 + 25 x + 29 y - 28 = 0. 

(6) x 3 - y 2 - 10 x 2 + 33 x - 36 = 0. 



132. Curve tracing. Some of the matters involved in curve 
tracing have been discussed in Arts. 75-78, 125-131. To do more 
than this is beyond the scope of a primary text-book on the 
calculus. The topic is mentioned here merely for the purpose of 
giving a few exercises whose solutions require the simultaneous 
application of methods for finding points of maximum and mini- 
mum, asymptotes, and singular points. 



212 DIFFERENTIAL CALCULUS. [Ch. XIII. 

Note 1. For a fuller elementary treatment of singular points and curve 
tracing, see McMahon and Snyder, Biff. Cat., Chaps. XVII., XVIII., 
pp. 275-306; F. G. Taylor, Calculus, Chaps. XVIL, XVIII., pp. 250-278; 
Edwards, Treatise on Diff. Cal., "Chaps. IX., XII., XIII.; Echols, Calculus, 
Chaps. XV., XXXI., pp. 147-164, 329-346. The classic English work on the 
subject is Frost's Curve Tracing (Macmillan & Co.), a treatise which is 
highly praised both from the theoretical and the practical point of view.* 

Note 2. For the application of the calculus to the study of surfaces (their 
tangent lines and planes, curvature, envelopes, etc.) and curves in space, see 
Echols, Calculus, Chaps. XXXII. -XXXV., pp. 347-390, and the treatises of 
W. S. Aldis and C. Smith on Solid Geometry. 

EXAMPLES. 

1. Trace the curves in Ex. 8, Art. 160; in Ex. 5, Art. 161; in Ex. 2, 
Art. 162 ; in Ex. 3, Art. 165. 

2. Trace the following curves : 

(1) y2 = x*(l - x" 2 ). (2) y 2 = x 2 (l~x). (3) x*-4x 2 y-2xy 2 + 4y 2 = 0. 
(4) 2 y 2 = 4 xy — x 3 . (5) r = a cos 4 0. 

133. NOTE SUPPLEMENTARY TO ART. 127. 

(In this Note parts of Exs. 6, 7, Art. 127, are worked. Figures should be 
drawn by the student.) 

Ex. 6. Find the asymptotes of the hyperbola 

b 2 x 2 - a 2 y 2 = a?W (1) 

by method (a) Art. 127. 

The equation of the tangent at a point P(x\, y{) on (1) is (Art. 61) 

a 2 Vi 
Hence the ^-intercept of the tangent 

_ 6 2 xi 2 - a 2 y? _ aW _ a 2 . , g . 

b 2 x± b 2 Xi X\ 

and the ^/-intercept of the tangent 

_ a 2 y x 2 - b 2 x x 2 _ a 2 b 2 _ b 2 „. 



* A recent important work on curves is Loria's Special Plane Curves, a 
German translation of which (xxi. + 744 pp.) is published by B. G. Teubner, 
Leipzig. 



132, 133.] . SINGULAR POINTS. 213 

When the point P(#i, y{) recedes to an infinite distance along the hyper- 
bola, Xi and y 1 each increases beyond all bounds. Accordingly the intercepts 
in (3) and (1) both approach zero as a limit. Hence a tangent which touches 
the hyperbola (1) at an infinitely distant point passes through the origin. 
The equation of the line through the origin (0, 0) and P(xi, y{) is 

y = y i. (5) 

x Xi ' ~ 

If line (2) is an asymptote, it passes through the origin ; substitution of 
(0, 0) and solution for ^1 gives 

El 

fe=±», (6) 

Xi a 

.-. from (5) and (6) the equations of the asymptotes of the hyperbola are 

y = ±-x. 
Ex. 7 . Examine for asymptotes the parabola 

2/ 2 = ipX, (7) 

by method (&), Art, 127. 

The equation of the tangent at a point P(xi, y{) on (7) is (Art. 61) 

y-yi=^(x-xi). (8) 

By analytic geometry, the length of the perpendicular from a point (h, k) to 
a line ax + by -f c = is 

ah + bk + c 



V«2 + p 
length of perpendicular from the origin (0, 0) on the tangent (8) 
, 2px 

-yi + - JL - n 

Vi _ 2px 1 — y 1 2 



A /i I £?! ^yi 2 + ±p 2 
x yi 2 

Since y± 2 = 4pxi, this reduces to 

2pxi _ ^P-x 1 = ^l™i_ , (9) 

2 Vp Vari+p Vxi+p /]_ + P. 

When the point P(x^ ?/i) recedes to an infinite distance along the pa- 
rabola, x.i increases beyond all bounds. Hence, length (0) increases beyond 
all bounds. Accordingly, the tangent which touches parabola (7) at an in- 
finitely distant point is itself at an infinite distance from the origin, and thus 
is not an asymptote. 



CHAPTER XIV. 



APPLICATIONS TO MOTION. PRELIMINARY NOTE. 



134. Speed, displacement, velocity. Suppose a point moves from 
to P, through, a distance As, in a time A£, either along a 
straight line or along any curve (Figs. 70, 71). 




Fig. 70. 



Fig. 71. 



The mean speed of the moving point during the time At = 

As 

The speed of the moving point at any instant* = lim A< ^ — 

_ds 
~ ' dt' 

(This has been shown in Art. 25.) 

The rate of change of speed = — (speed) = — (■ 

az a i \ at 



As 

A*' 



Displacement. 




Fig. 72. 



= d2s 
dtf' 

If a point moves from one point to another, no 
matter by what path, its change 
of position (only its original and 
final positions and no intermedi- 
ate position being considered) is 
called its displacement. 

According to this definition, 
if a point moves from P to P x 
along any path PAP 1} say, its 
Y displacement is known com- 
pletely when the length and 



* One may also say the speed of the moving point at any point in its path. 

214 



134.] APPLICATIONS TO MOTION. 215 

direction of the straight line PP X are known. A displacement 
thus involves both distance and direction. The length of the 
line PP X is called the magnitude of the displacement ; the direc- 
tion of the line PP 1 is called the direction of the displacement. 
Thus the straight line PP X represents the displacement which a 
point has when its position shifts from P to P v 

Mean Telocity. Telocity. The mean velocity of a moving point 1 
which has a certain displacement in a time At J 

_ its displacement in time At 

"" At 

Thns the mean velocity, since it depends on a displacement, 
takes account of direction. E.g. in Fig. 72, if a point moves along 
the curve from P to P l in a time At, 



its mean speed 



are PAP, 

At ' 



., , ., chord PP. 

its mean velocity = * • 

J At 

That is, on denoting the arc and the chord in Fig. 72 by As and 
Ac, respectively, 

mean speed = — ; (1) 

mean velocity = (2) 



Tlie velocity of a moving point at any instant 



displacement /Q \ 

At 

This velocity can be represented by the displacement that 
would be made in a unit of time were the velocity to remain 
unchanged during that time (or remain uniform, as it is termed). 
From the above definitions it follows that : 

speed involves merely distance and time ; 

velocity involves direction as well as distance and time. 

* One may also say the velocity at any point. 



216 DIFFERENTIAL CALCULUS. [Ch. XIV. 

135. To find for any instant (or at any point) the velocity of 
a point which is moving along a curve. It has been shown in 
Art. 134, result (2) (see Fig. 72), that when a point moves along 
the curve from P to P 1} 

its mean velocity = — ■ 

J -A* 

Ac 
Now, velocity at P — lim A , i0 — 

-, • /Ac As' 



Ac r As 

As Af 



* 



= 1 ■ * [See Arts. 25, 67 (c), (d).] 

_ds 
"~'d«" 

Thus the magnitude of the velocity at P is the same as the 
magnitude of the speed at P. The direction of the velocity at 
P is the same as the direction of the tangent at P; since the 
chord PP X approaches the tangent as its limiting position when 
A£ = 0. 

Note. Velocity may change owing to a change in the direction of motion, 
or to a change in speed, or to changes in "both direction and speed. Thus 
the velocity of a point moving in a straight line with ever increasing speed is 
changing ; the velocity of a body moving in a circle with uniform speed is 
changing ; the velocity of a hody moving with changing speed along any 
curve is changing. 

136. Composition of displacements. Suppose a particle has 
successively the displacements a and b. 




A 
Fig. 74. 



* As is not zero when At is not zero. 



135, 136.] APPLICATIONS TO MOTION. 217 

The resultant of these two displacements can be shown thus : 
Through any point draw OA parallel and equal to a ; through 
A draw AB parallel and equal to b. A particle which, starting at 
0, undergoes successively the displacements a and b, must arrive 
at B. The particle would also have arrived at B, if, instead of 
having these displacements, it had the displacement represented 
by OB. The displacement OB (or a displacement equal and 
parallel to OB) is called accordingly the resultant of the displace- 
ments a and b. 

Fig. 74 shows that "if two sides of a triangle taken the same 
way round represent the two successive displacements of a moving 
point, the third side taken the opposite way round will represent 
the resultant displacement." 

When there are more than two successive displacements, the 
resultant is obtained in a manner similar to the above. Thus, 
for example, let a, b, c, represent three successive displacements 
of a moving point. 




Through any point draw OA parallel and equal to a, through 
A draw AB parallel and equal to b, through B draw BG parallel 
and equal to c. A particle which, starting at 0, undergoes suc- 
cessively the displacements a, b, c, must arrive at C. The particle 
would also have arrived at C, if instead of having these displace- 
ments it had the displacement represented by OC. The single 
displacement OC (or a displacement equal and parallel to OC) is 
accordingly called the resultant of the displacements a, b, c. The 
resultant of any finite number of displacements can be found by 
an extended use of the methods used in the preceding cases. 

EXAMPLES 

1. A point undergoes two displacements, 40 ft. E. and 30 ft. N. Find 
the resultant displacement. 



218 



DIFFERENTIAL CALCULUS. 



[Ch. XIV 



2. A point undergoes two displacements, 60 ft. W. 30° S. and 30 ft. N. 
Find the resultant displacement. 

3. A point undergoes three displacements, 12 ft. W., 20 ft. N. W., and 
60 ft. N. E. Find the resultant displacement. 

4. To an observer in a balloon his starting point bears N. 20° E., and is 
depressed 30° below the horizontal plane ; while a place known to be on the 
same level as the starting point and 10 miles from it is seen to be vertically 
below him. Find the component displacements of the balloon in southerly, 
westerly, and upward directions. 

137. Resolution of a displacement into components. A displace- 
ment can be resolved into component displacements (or, briefly, 
components) which have that displacement as their resultant. 
This may be done in an unlimited number of ways. For instance, 
in Figs. 76, 77, 78, various pairs of components (in light lines) 
are shown for the displacement a. 





Fig, 76. 



Fig. 77. 



Fig. 78. 



The components are often represented by drawing them from 
O ; thus corresponding to Figs. 76, 77, 78, are Figs. 79, 80, 81, 
respectively. 




Fig. 79. 



>P 




>P 



Fig. 80. 



Fig. 81. 



Components which are at right angles to one another, like 
those shown in Figs. 78, 81, are called rectangular components. 

If a displacement a is inclined at an angle to its horizontal 
projection, the horizontal and vertical coynponents of the displace- 
ment (as is evident from Figs. 78, 81) are respectively 



a eos 0, a sin 0. 



136, 138.] APPLICATIONS TO MOTION. 219 

EXAMPLES. 

1. A particle has a displacement of 12 feet in a direction making an 
angle of 35° with the horizon. What are the horizontal and vertical com- 
ponents of the displacement ? 

2. The vertical component of a displacement of 35 ft. is 24 ft. Find the 
horizontal component and the direction of the displacement. 

3. The horizontal component of a displacement is 300 ft., and the direc- 
tion of the displacement is inclined 37° 20' to the horizon. Find the ver- 
tical component of the displacement and the displacement itself. 

4. One component of a displacement of 162 ft. is a displacement of 236 ft. 
inclined at the angle 78° 40' to the given displacement. Find the other 
component. 

138. Composition and resolution of velocities. It has been re- 
marked in Art. 134 that the velocity of a moving particle at any 
instant may be represented by the displacement which the parti- 
cle would have in a unit of time were the velocity to become and 
remain uniform. Accordingly, velocities may be combined, and 
may be resolved into components, in precisely the same manner 
as displacements (Arts. 136, 137). 

EXAMPLES. 

1. A book is moved along a table in an easterly direction at the rate of 2 
ft. a second ; at the same time the table is moved across the floor at the rate 
of 1 ft. a second in a southerly direction. Find the resultant velocity of the 
book with respect to the floor. 

2. A steamer is going in a direction N. 37° E. at the rate of 18 miles per 
hour, and a man is walking on the deck in a direction N. 74° E. at a rate of 
3 miles per hour. Find the resultant velocity of the man over the sea. 

3. A river one mile broad is running at the rate of 4 miles per hour, and 
a steamer which can make 8 miles per hour in still water is to go straight 
across. In what direction must she be steered ? 

4. A man is driving at a rate of 12 miles per hour in a direction N. 
18° 40' E. Find the rate at which he is proceeding towards the north and 
towards the east respectively. 

5. A train is running in the direction S. 48° 17' W. at a rate of 32.4 miles 
per hour. Find the rates at which it is changing its latitude and longitude 
respectively. 



220 



DIFFERENTIAL CALCULUS. 



[Ch. XIV. 



139. Component velocities of a point moving along a curve. Let 

the rectangular and polar 
T coordinates of the point be 
as in Fig. 82. 

(a) Components parallel to 
the axes. It has been seen 
in Art. 135 that the velocity 
v of the moving point when 
it is passing through P has 
the direction of the tangent 
at P and that in magnitude 

_ds 

Fig. 82. dt 

When the point moves, its abscissa and coordinate generally 
change. 

The rate of change of the abscissa x = — ; 
8 dt 9 

the rate of change of the ordinate y = -^ • 
8 " dt 

These are the components of v along the axes ; and thus 




/ ds\ 2 _ / docy I dy\^ # 



v^* 



\dt) 



dt 



(1) 



If the direction of motion PT makes an angle a with the a>axis, 



dx 
dt 



v cos a, 



dy 
dt 



= v sm a. 



(6) Components along, and at right 
angles to, the radius vector. 

In Fig. 82, x = r cos 0, y = r sin 0. 
.'., on differentiation, 



dx n dr ■ n dO 

— = cos r smO — 

dt dt dt 

dy . n dr . n d0 

-+ = smd h r cos — 

dt dt dt 



(2) 




139.] 



APPLICATIONS TO MOTION. 



221 



Now, as is apparent from Fig. 84, 
vel. along radius vector OP= component 

of — - along it + component of — ^ along it 



dt 



dt 



dx n , dy • 

— X cos 8 + — sm 
dt dt 



dr 
dt 



[from (2) and (3)]. 





(B).o^ 



Similarly, it may be seen 
[Fig. 85] that 
vel. at right angles to radius vector 



dy 

= — cos 

dt 

dQ 



dx . n 
dt 



(5) 



Fig. 85. 



= r^ [from (2) and (5)]. (6) 

From (1) on the substitution 
of the values of ■— , -~ , from (2), or, directly from (4) and (6), 



d8\* = (d?\*,(rd*\* 
dt] \dt) \ dt) 



(7) 



Note. The equality of the second members of (3), (4), and the equality 
of the second members of (5), (6), can also be deduced from the relations 
(see Fig. 82) 

r 2 = x 2 + y 2 (8) ; 

For, from (8), on differentiation, 



x 



(9) 







dt 


dt 


+ yf t ; 

dt 






whence 




dt 


x dx 
r dt 


,ydy 






i.e. 




dr_ 
dt 


cos 8 


^ + sin 

dt 


dt 




Also, from 


(9) 


on differentiation 


, 












x f ^ 


y dt 


x^- 


dx 
dt 






dd 


dt 


dt 






dt' 


X* 


+ y 2 


r i 




whence 




dt 


_x dy 
' rdt 


y dx 
rdt 












-- cos 6 


^-sin 

dt 


d dx 

dt 





222 



DIFFERENTIA L CAL C UL US. 



[Ch. XIV. 



EXAMPLES. 

Note. See Examples, Art. 65. 

1. A point is moving away from the cusp along the first quadrant branch 
of the curve y 2 = x* at a uniform speed of 6 in. per second. Find the respec- 
tive rates at which its ordinate and abscissa are increasing when the moving 
point is passing through the point (4, 8). Also find the rate at which its dis- 
tance from the cusp is increasing. 

Since y 2 = x B , 




<] 'l 



dx 



2y^. = Sx 2 ^ 



at 



at every point on the curve. 
16 dy = 
dt 

dt 

2 



48 



dt 

Hence at (4, 8) 
dx 



dt' 
dx 
dt 



Also 



Nf) 



80. 



ds 
dt 



(1) 

(2) 



Fig. 86. 



Also 
in which 



On solving (1) and (2), 

— = 1.897 in. per second ; 
dt 

dr dx 

— =. — t 

dt dt 
6 = tan- 1 1 = tan- 1 2. 



-f^sin 
dt 



dr 
dt 



1.897 x _i- + 5.69 x^ 

V5 V5 



: 5.69 in. per second. 

[Eqs. (3), (4).] 
(See Fig. 86.) 
5.94 in. per second. 



2. In each of Exs. 1, 2, Art. 65, find the rate at which the moving particle 
is increasing its distance from the vertex of the parabola. 

3. In each case in Exs. 3, 5, Art. 65, find the rate at which the moving 
particle is increasing its distance from the origin of coordinates. 

4. The radius vector in the cardioid r = a(l — cos 0) revolves at a uniform 
rate about the pole : investigate the motion of the point at the extremity of 
the radius vector. Apply the results to determining the motion of this point 
at the following points on the cardioid in which a = 10 inches, when the 
radius vector makes a complete revolution in 12 sec, viz. at the points 



(1) 



(lO, |); (2) (5, !); (3) (l5, ?Z); (4) (20, t) 



[Suggestion. Find (a) the velocity of the moving point toward or away 
from the pole ; (6) the velocity of the moving point at right angles to the 
radius vector ; (c) the velocity of the moving point along the cardioid.] 



139, 140.] APPLICATIONS TO MOTION. 223 

140. Acceleration. The rate at which a body is moving may 
change, either becoming greater or becoming less ; the direction 
of its motion may also change ; again, both the rate and the direc- 
tion of its motion may change. 

E.g. a train may be moving at one instant at a rate of 10 miles per hour ; 
ten minutes later it may be moving at a rate of 40 miles an hour. The rate 
at which the train moves has thus increased by 30 miles an hour in ten 
minutes. 

The change made during an interval of time in the velocity of 
a body is called the total acceleration, and also the integral acceler- 
ation for that interval. Thus, suppose (Fig. 87 a) a body at one 




v 2 



Fig. 87 a. Fig. 87 b. 

moment has a velocity v ly and at another moment some time later 

has a velocity v 2 . Fig. 87 6 shows that the velocity v 2 can be 

obtained by compounding the velocity AB with the velocity v v 

Thus AB represents the change that must be made in the velocity 

v 1 in order that the velocity of the body may become v 2 . In this 

instance AB is called the integral, or total, acceleration of the body. 

The mean (or average) acceleration of a body is the result obtained 

by dividing the integral acceleration by the number of units of 

time that has elapsed while the integral acceleration was in the 

making. Thus if (Figs. 87a, b) v 1 changed to v 2 during an interval 

of t seconds, . 

the mean acceleration = 

t 

This may be called the change in the velocity per unit of time. 
The direction of the mean acceleration is the direction of the 
integral acceleration. 

The instantaneous acceleration of a moving point at any moment, 
usually called ' the acceleration? is the limit, in magnitude and 
direction, of the mean acceleration when the interval of time, t, is 
taken as approaching zero. The acceleration is usually denoted 
by the letter a. 



224 DIFFERENTIAL CALCULUS. [Ch. XIV. 

In symbols : if the velocity v has a change Av in a time At, 

the acceleration = lim A<:M) — : 

At 



i.e. a =^r» (1) 



dv 

at 



Accelerations have direction and magnitude ; accordingly, they 
can be represented by straight lines. Accelerations may be com- 
bined and may be resolved into components, in precisely the same 
way as displacements and velocities. 

Note. Another form for the acceleration a is 

„ _ dv _ dv ds _ m dv /0 x 

(Mi — — — — - • — - — V — — • {&) 

dt ds dt ds 

EXAMPLES. 

1. The initial and final velocities of a moving point during an interval of 
3 hours are 20 miles per hour W. and 16 miles per hour N. 43° W. Find 
(a) the integral and (6) the mean acceleration. Also find the easterly and 
northerly components of these accelerations. 

2. A particle is moving downwards in a direction making 36° with the 
vertical, and the vertical component of its acceleration is 80 ft. per second 
per second. Find (a) acceleration in the path of motion and (b) the hori- 
zontal component of its acceleration. 

141. Acceleration : particular cases. 

(a) Acceleration of a point moving in a straight line. 

By Art. 140, (1) a = — • 

Now v = — ; 

dt' 

d , v d fds\ d 2 s ,--. N 

Note. In the case of a point that is moving on a curve, the direction of 
the velocity at any point of the curve is along the tangent at that point and 

rjo /72o 

the velocity (Art. 135) is — Accordingly in this case — represents merely 
dt dt 2 

the acceleration of the moving point in the direction of the tangent, the 

tangential acceleration, as it is termed. This is also shown in (5) following. 



140, 141.] 



APPLICATIONS TO MOTION. 



225 



EXAMPLES. 

1. In the case of a body falling vertically from rest, the distance s fallen 
through in t seconds is given by the formula s = \ gt 2 . Show that the accel- 
eration is g. 

2. A point P is moving at a uniform rate round a vertical circle. An 
ordinate PM is drawn to meet the horizontal diameter in M. Find the 
acceleration of M with respect to the centre of the circle. 

3. Suppose that the circle in Ex. 2 has a radius 3 ft. and that P goes 
round the circle 25 times per second. Find the acceleration of M : (a) when 
P is 20° above the horizon ; (6) when P is Qb° above the horizon. 




(6) Acceleration of a point moving in a plane curve.* In order to 
determine this acceleration at any point two rectangular components 
of it are first found ; namely, the 
acceleration along the tangent 
at the point and the acceleration 
along the normal. These are 
called the tangential and the nor- 
mal accelerations. 

Suppose a point moves along 
the curve in Fig. 88 from P x to 
P 2 in a time A£, and let its veloci- 
ties at P 1 and P 2 be v and v + Ay, 
respectively. 

Let PiRi and P 2 R 2 represent these velocities in magnitude 
and direction. 

Draw P^S equal and parallel to P 2 R 2 and join R X S. 

Then R^ represents in magnitude and direction the change in 
velocity, Av, made during the time At. From S draw SQ at right 
angles to PjQ, the normal at P 1? and draw ST at right angles to 
Pi ^ T, the tangent at P v Denote the arc PiP 2 by As, and the 
angle between P 1 R 1 and P 2 R 2 (i.e. angle TP^) by A<£. 

Denote the tangential acceleration by a t , and the normal accel- 
eration by a n . The components of P^, in the directions of the 
tangent and normal at P 1? respectively, are R X T and T/S, the 
latter of which is equal to PiQ. 



* See Campbell's Calculus, Art. 25/ 



226 
Then 



DIFFEBEN TIAL CALCUL US. 



[Ch. XIV, 



a t = lini z 



= lim A ^ 
= lim AfcM) 



A£ 



= 11111, 



= limy 



"PiScosAft-P, 



^f] 



A£ 

(i; -f- A^) cos Aft — g 
A* 



i 1 (cos Aft — 1) + Aw cos Aft~ | 

A£ J 



z v sin 



2 1 



Aft 



At 



+ 



^ cos Aft] 



" sin \ Aft ( — v sin ^ A ft) Aft Aw 



|Aft 



^ A* 



cos A ft 



= 1.0.** + ^ 

dt 



dv 
dt 



dt 
d*s 



(2) 



Further a n = liin AfrM) 



= lim, 



PiQ_ 



At 



lim 



(<y + A<y) 



a*m>— = lim A „ 

A£ 

sill Aft Aft As' 
Aft As A? 
tfft 



P^ sin Aft 

1 A* 



ds dt ds 



ds 

dt 

# = 1, Arts. 98-101 
c?s r 

in which r denotes the radius of curvature at the point. 
.-. the actual, or resultant, acceleration 



(3) 



=4 



cfs\ 2 ^ 

dt 2 ) ~V" 

Special case. When a point is moving uniformly in a circle, 
there is no tangential acceleration. The acceleration at any point 

is then wholly directed towards the centre and its magnitude is — . 

Ex. Show that when a point moving with uniform speed goes 
round a circle of radius r in time t, its acceleration at any instant 

has the magnitude ■ - • 



141.] APPLICATIONS TO MOTION. 227 

EXAMPLES. 

4. A circus rider is moving with the uniform speed of a mile in 2 min. 
40 sec. round a ring of 100 ft. radius: find his acceleration towards the centre. 

5. A point moving in a circular path, of radius 8 in., has at a given posi- 
tion a speed of 4 in. per second which is changing at the rate of 6 in. per 
second per second. Find (a) the tangential acceleration; (6) the normal 
acceleration ; (c) the resultant acceleration. 

6. A particle is moving along a parabola y 2 =4 x, the latus rectum of 
which is 4 inches in length, and when it is passing through the point P (4, 4) 
its speed, which is there 6 in. per second, is increasing at the rate of 2 in. per 
second per second. Find at P, (a) its tangential acceleration ; (5) its normal 
acceleration ; (c) its integral acceleration. 

7. If the particle in Ex. 6 were moving at a uniform rate of 6 in. per 
second, what would be its acceleration at P? 

Xote 1. When a point is moving along a curve, the coordinates x, y 
of its position are continually changing. The components of its acceleration 
at P (x, y) which are parallel to the x and y axes are respectively [compare 
Art. 139 («)] ^ ^ 

at* dt 2 ' 

If the tangent to the curve at P makes an angle a with the x-axis, then, 
as is apparent from a figure, the tangential acceleration 

(4) 







^ = ^cosa+^sina 
dt 1 dt 2 dt 2 


Eelation 


(4) 


follows also from result (1) Art. 139 (< 
(ds\2_ ldx\ 2 (dy\ 2 
{dt) ~ [dt] \dt) ' 


For, on differentiation, 






ds _ d 2 s _ dx m d?x dy > d 2 y . 
dt ' dt 2 ~ dt ' dt 2 dt ' dt 2 ' 


whence 




d 2 s _ dx d 2 x dy # d 2 y 
dt 2 ~ ds ' dt 2 ds ' dt 2 ' 


i.e. 




^=cosa. ^+sina. 
dt 2 dt 2 



(5) 



(6) 



my 

dt 2 

Note 2. Angular Telocity. Angular acceleration. The mean rate at 
which a straight line revolves about a given point (i.e. mean rate at which 
it describes an angle from a certain initial position) is called the mean angu- 
lar velocity of revolution. 

E.g. if a straight line revolving about a point describes the angle - in 

o 

4 sees. , its mean angular velocity per second is — -h 4, i. e. — radians per second. 



228 



DIFFERENTIAL CALCULUS. 



[Ch. XIV. 



The instantaneous angular velocity, commonly called the angular veloc- 
ity, at a particular moment, Ad denoting the angle described in a time At, 

_ ,. Ad dQ 

= Iim A ^o — - = — • 
At dt 

The angular acceleration at any moment is the rate of change of the 
angular velocity. Accordingly, 

d fdd\ d 2 Q 



angular acceleration = - (—) = ~ 

dt\dt) dt 2 



(7) 



EXAMPLES. 

8. A wheel is rolled at a uniform rate along a straight line; investigate 
the motion of a fixed particular point P on the wheel. 

The particular point P on the wheel describes a cycloid. If the axes be 
chosen in the usual way, the equations of the cycloid are 

x = a (0 — sin 0) 1 
y = a (1 — cos 0) J 

in which a denotes the radius of the wheel and denotes the angle through 
which the radius through P turns after P has been on the straight line. 



(8) 




/L 10 



It is required to investigate the motion of the point P of the wheel at any 

point on its cycloidal path. 

d6 
Since the wheel is rolling at a uniform rate, — has a constant value and 

dt 

accordingly — = 0. 
dt 2 

In Fig. 89 PT is the tangent to the cycloid at P, and PN is the normal. 

From (8), on differentiation, 



*? = fl(l-COS*)^ 

dt dt 
dt dt 


dt 2 \dt) 

S l!h ^ = acos*W 
dt 2 \dt) 


Hence, on substitution in Art. 139, Eq. 1, 


velocity v at 


P = ( ^ = 2asm d - . c ™. 
dt 2 dt 



(10) 



(11) 



141.] APPLICATIONS TO MOTION. 229 

From (11), on differentiation, and Art. 141, Eq. 2. 

the tangential acceleration at P, a t = — = acos-( — ] . (\2) 

dV l 2\dt) y J 

Result (12) can also be derived from Eq. (5), Note 1, on substitution of 
the values of the derivatives from (9), (10), (11), above. 

Result (12) can also be derived from Eq. (4), Note 1, on observing that the 

a 

tangent PT makes an angle 90 with the x-axis. 

The radius of curvature r at P [Art. 101, Ex. 5 (8)] = 4 a sin -• (13) 

Hence by Art. 141 and Eqs. (11) and (13) above, 

v 2 2\dt 

the normal acceleration at P, a n = — = - — '— 

r a 

4 a sin - 

—1(f)- M 

. •. integral acceleration at P = Va/ 2 + a~ n l = a [ — V (15) 

\dt I 

On making a figure showing the accelerations ( 12) and (14), which are 
directed along PT and PN respectively, it will be apparent that acceler- 

a 

ation (12) makes an angle - with the resultant acceleration. Accordingly, 

the resultant acceleration of the point on the wheel at any point on its 
cycloidal path is constant, and is always directed towards the centre of the 
wheel. 

9. Suppose the wheel in Ex. 8 has radius 2 feet, and is pushed along at a 
rate of 3 miles an hour. Calculate the velocity and the tangential, normal, 
and integral accelerations of a point on the wheel the radius to which makes 
an angle of 60° with the vertical radius downward from the centre. 

10. If the wheel in Ex. 8 is not rolling at a uniform rate, show in each of 
the three ways indicated for deriving result (12) in that example, that the 
tangential acceleration at P is 

2 ffl sin^ +(I cos^^ 2 
2 df- 2\dt, 



CHAPTER XV. 

INFINITE SERIES. 

EXPANSION OF FUNCTIONS IN INFINITE SERIES. DIFFEREN- 
TIATION OF INFINITE SERIES. SERIES OBTAINED BY 
DIFFERENTIATION. 

N.B. There are some students whose time is limited and who require to 
obtain as speedily as may be a working knowledge of Taylor's and Mac- 
laurin's expansions. These students had better proceed at once to Arts. 149, 
154, work the examples in Arts. 150 and 152, and then take up Art. 148. 
It is, perhaps, advisable in any case to do this before reading this chapter and 
the other articles in Chapter XVI. Those who are studying the calculus as 
a "culture " subject should become acquainted with the ideas and principles 
described, or referred to, in Chapters XV., XVI. A thorough understand- 
ing of these ideas and principles is absolutely essential for any one who 
intends to enter upon the study of higher mathematics. 

142. Infinite series : definitions, notation. An infinite series 
consists of a set of quantities, infinite in number, which are con- 
nected by the signs of addition and subtraction, and which suc- 
ceed one another according to some law. A few infinite series of 
a simple kind occur in elementary arithmetic and algebra. 

For instance, the geometrical series 

1 + H+™ + »H+5: + i5» + '-' (1) 

the geometrical series 

1 +£+ Z 2 + ... + X n ~ 1 + £« + £»+l + ..., (2) 

which may also be obtained by performing the division indicated in ; 

the geometrical series 1 — x 

l-as + a?+...+(-l)«a?-i + -, 

which may also be obtained by performing the division indicated in 
the geometrical series 



1 



1+B 

a + ar + ar 2 + ••• + ar"- 1 + ar n + ar n+l + ••• ; (4) 

the series I4.J_4.J__) !- — +•••. (5) 

\p 2p Hp nP 

230 



142, 143.] INFINITE SEBIES. 231 

The successive quantities in an infinite series, beginning with 
the first quantity, are usually denoted by 

u , Ujj u 2 , •••, u n -i, u n , u n+ i, •••; 

or, in order to show a variable, x say, by 

u (x), u^x), u 2 (x), •, «»_!(«), u n (x), u n+ i(x), .... 

Then the series is 

Mo + m x + « 2 H h m„_i + w n + w M+ i H . (6) 

The value of the series is often denoted by s ; and the symbol s n 
is generally used to denote the sum or value of the series obtained 
by taking the first n terms of the infinite series; thus, 

s n = u -f % + u 2 H h M n _i. 

The value of the infinite series (6) is the limit of the sum of the 
quantities in the series; i.e. the value of the series is the limit of 
the sum of n terms of the series when n increases beyond all 
bounds.* This is expressed in mathematical symbols 

s = lim„ ico s n . (7) 

(This limit s is frequently, but not quite correctly, called " the 
sum of the series" or "the sum of the series to infinity") 

Thus, in (1), s„= 1 +1 + 1+ ... + JL = 2^1 - -1A 

and hence s = lini,^^ s n = 2 ; (7) 

in (2), s n = 1 + x + x 2 + ... + a?*- 1 = *^i 

x — 1 

and hence s = lirn^xSrc = co when x-^.1 and xS — 1, (8) 

= — — when - 1< x< 1. (9) 



143. Questions concerning infinite series. The subject of infinite 
series is highly important in mathematics. Such questions as the 
following arise and require to be answered : 

(a) Under what conditions may infinite series be employed in 
mathematical investigation and used in practical work ? 

* Thus s is not the sum of an infinite number of terms of the series, but is 
the limiting value of that sum. 



232 DIFFERENTIAL CALCULUS. [Ch. XV. 

(6) Under what conditions may an infinite series be used to 
define a function or employed to represent a function ? 

Thus, in Art. 167, result (8) shows that series (2) does not represent the 

function ■ when x is greater than 1 or less than — 1 or equal to 1 or — 1. 

1 — x 

This is obvious on a glance at the series ; in fact, the greater the number of 
terms of (2) that are taken, the greater is the error committed in taking the 
series to represent the function. (For instance, put x = 2 ; then the func- 
tion is — 1 and the series is + oo.) On the other hand, the infinite series (2) 

does represent the function when x lies between — 1 and + 1 ; the 

1 — x 

greater the number of terms that are taken, the more nearly will the sum of 

these terms come to the value of the function. The limit of the sum of these 

terms when the number of them is infinite is the function. 

(c) May two infinite series be added like two finite series ? In 

other words, if 

u = u + u 1 + u 2 -\ 

and v = v Q + v 1 -\- v 2 + •••, 

is u-\-v = u + v +u l + v 1 -\ (1) 

a true equation; and under what conditions is (1) a true equation? 

(d) May two infinite series be multiplied together like two 
finite series ? In other words, u and v being as in (c), is 

uv = u v + u v 1 + UjVq + u 1 v 1 + u^v 2 + u 2 v x + ••• (2) 

a true equation; and under what conditions is (2) a true equation ? 

(e) May the principles of Art. 31 and Art. 174 A, namely, that 
the derivative and the integral of the sum of a finite number of 
terms are respectively equal to the sum of the derivatives and the 
sum of the integrals of these terms (to a constant), be extended 
to infinite series ? That is, u 0) u 1} u 2 , •••, being functions of x, if 

s = u + Wj + ^H , 

Jsdx = I v dx + | U\dx -\- I u 2 dx -f- • • •, (3) 



are 



and i-(.) =J-0„) +-p(«d +!"(%) + -, ( 4 ) 

dx dx dx dx 



143. 144.] INFINITE SERIES. 233 

true equations; and what are the conditions which must be 
satisfied in order that these equations be true ? Equations (3) and 
(4) may be expressed : 

J lim,^ s n (x) dx = lirn^ j s n (x)dx , 
|[lim_ <.(»)] = lim_[| S „(,)]. 

The above questions then may be stated thus : Is the integral 
of the limit of the sum of an infinite number of quantities equal to 
the limit of the sum of the integrals of the quantities ; and is it 
likewise in the case of the differentials ? 

For instance, given that = 1 + x + x 2 4- X s + •••, 

1 — x 

and " ££-,!>* los rrj = " + f + f + - ? 

144. Study of infinite series. Knowledge, elementary knowledge at 
least, of the theory of infinite series, and practice in their use are necessary in 
applied mathematics. Infinite series frequently present themselves in the 
theory and applications of the calculus, and accordingly the subject should 
he studied, to some extent at least, in an introductory course in calculus. 
The better text-books on algebra, for instance, among others, Chrystal's 
Algebra (Vol. II., Ed. 1889, Chap. XXVI., etc.), Hall and Knight's Higher 
Algebra (Chap. XXI.), contain discussions on infinite series and examples for 
practice.* Osgood's pamphlet, Introduction to Infinite Series (71 pages, 
Harvard University Publications) , gives a simple, elementary, and excellent 
account of infinite series. "This pamphlet is designed to form a supplemen- 
tary chapter on Infinite Series to accompany the text-book used in the course 
in calculus." Becent text-books on the calculus, in particular those of 
McMahon and Snyder, Lamb, and Gibson, contain definitions and theorems 
on infinite series ; they will especially well repay consultation. More 
elaborate expositions of the properties of infinite series, which form parts of 
introductory courses in modern higher analysis, are given in Harkness and 
Morley, Introduction to the Theory of Analytic Functions, in particular 

* Also see Hobson, A Treatise on Plane Trigonometry, Chap. XIV. , and 

following chapters. 



234 DIFFERENTIAL CALCULUS. [Ch. XV. 

Chaps. VIII -XL, and in Whittaker, Modern Analysis, in particular Chaps. 
II.-VIII. These discussions can be read, in large part, by one who possesses 
a knowledge of merely elementary mathematics. 

A statement of a few of the principal definitions and theorems which are 
necessary for an elementary use of infinite series is given in Arts. 145-147. 

145. Definitions. Algebraic properties of infinite series. An 
infinite series has been defined in Art. 142. If (see Art. 142) 
lim,^^ s n is a definite finite quantity, U say, the series is called 
a convergent series, and is said to converge to the value U. If s n 
does not approach a definite finite value when n approaches 
infinity, the series is called a divergent series. In a divergent 
series, when n approaches infinity, s n may either approach infinity, 
or remain finite but approach no definite value. 

Thus, in Art. 142, series (1) is convergent ; series (2) is convergent for 

values of x between — 1 and + 1, for then s = ; series (4) is convergent 

1 —x 

when r lies between — 1 and -f 1, for then s = — - — Series (5) is con- 

1 — r 

vergent for^> > 1, and divergent forp = 1 and for p < 1. (Hall and Knight, 
Algebra, p. 235.) 

[Note 1. The harmonic series. When p = 1, series (5) is 

1+- + - + - + -+ ••• +- + — — + ■••■ 
23 45 n n + 1 

This series is called the harmonic series.'] 

The series 1 + 2 + 3 + 1-«+- is divergent. The series 1—1+1—1+ 

••• + (— l) n_1 + •••, obtained by putting x = 1 in series (3), is divergent ; for 
its limit is or 1 according as n is even or odd. (A series that behaves like 
this is said to oscillate. Some writers do not include oscillatory series among 
the divergent series.) 

In general only convergent series are regarded as of service in 
applied mathematics. (For the necessity of the qualifying phrase 
"in general," see Note 2.) A series may be employed to represent 
a function, or, what comes to the same thing, a function may be 
defined by a series, if the series is convergent. Thus series (2), 

Art. 142, may be used to represent or to define , if x lies 

JL — X 

between — 1 and + 1. [See questions (a) and (&), Art. 143.*] 

* Carl Friedrich Gauss (1777-1855), the great mathematician and astrono- 
mer of Gottingen, and Augustin-Louis Cauchy (1789-1857), professor at the 



145.] INFINITE SERIES. 235 

Note 2. On divergent series. Those who apply mathematics, astrono- 
mers in particular, have frequently obtained sufficiently good approximations 
to true results by means of divergent series. Such series, however, " cannot, 
except in special cases, and under special precautions, be employed in mathe- 
matical reasoning" (Chrystal, Algebra, Vol. II., p. 102). At the present 
time considerable attention is being paid by mathematicians to divergent 
series and to investigations of the fundamental operations of algebra and the 
calculus upon them. A work on the subject has recently appeared, viz. 
Lemons sur les series divergentes, par Emile Borel (Paris, Gauthier-Villars, 
1901, pp. vi + 182). "It is safe to say that no previous book upon diver- 
gent series has ever been written." Interesting and instructive information 
concerning divergent series will be found in reviews on this book, by G. B. 
Mathews {Nature, Nov. 7, 1901), and E. B. Van Vleck (Science, March 28, 
1902). 

Absolutely convergent series. A series the absolute values (see 
Art. 8, ISTote 1) of whose terms make a convergent series is said 
to be absolutely or unconditionally convergent; other convergent 
series are said to be conditionally convergent. 

Ex. 1. Series (1), Art. 142, is an absolutely convergent series. 

Ex.2. The series !-£+$-£+$ («) 

may be written (1 - i) + (i - i) + Q - i)+ ..., i.e. i + T ^ + ^-f- .... 

Series (a) may also be written 

i -(*-*)-(*-*)-. «■«■ i-*-A- — 

Thus the value of the series (a) , the terms being taken in the order indi- 
cated, is less than 1 and greater than i It can also be shown that this series 
converges to a definite value. On the other hand (see Note 1, and the state- 
ment just preceding Note 1), the series 

is divergent. Thus series (a) is a conditionally convergent series. 

Theorems. (1) If a series is absolutely convergent, it is obvious 
that any series formed from it by changing the signs of any of 
the terms is also convergent. 

Polytechnic School at Paris, who did much to make mathematics more rigor- 
ous than it had been during its rapid development in the eighteenth century, 
may be regarded as the founders of the modern theory of convergent series. 
James Gregory, professor of mathematics at Edinburgh, introduced the terms 
convergent and divergent in connection with infinite series in 1668. 



236 DIFFERENTIAL CALCULUS. [Ch. XV. 

(2) In a conditionally convergent series it is possible to rearrange 
the terms so that the new series will converge toward an arbitrary 
preassigned valne. 

(3) In an absolutely convergent series the terms can be rearranged 
at pleasure without altering the value of the series. 

(4) If (see Art. 143) u and v are any two convergent series, they 
can be added term by term ; that is, Equation (1), Art. 143, is true. 

(5) If u and v are any two absolutely convergent series, they 
can be multiplied together like sums of a finite number of quanti- 
ties ; that is, Equation (2), Art. 143, is true. 

For proofs and examples of these theorems see Osgood, Intro- 
duction to Infinite Series, Arts. 34, 35 ; Chrystal, Algebra, Yol. II., 
Chap. XXVI. , §§ 12-14. 

In a convergent series as n increases, s n may either: (a) con- 
tinually increase toward the limiting value of the series ; or 
(b) decrease toward this limit ; or (c) be alternately greater than 
and less than its limit. 

Thus in series (1), Art. 142, s n continually increases toward its limit (2); 

in the series 1 1 \- •••, s n is alternately greater than and less than 

its limit f. 2 22 23 

Remainder after ft terms. The symbol r n or M n is often used to 
denote the series (and also to denote the value of the series) 
formed by taking the terms after the nth, thus 

^ = ™* + W w+1 + ^n+2 H • 

This is usually called the remainder after n terms. Let a func- 
tion be represented by a convergent series ; i.e. let the value of 
the function be equivalent to the value of this convergent series. 

Then since ,, . ,. ,. 

the tunction = lim ni00 s n> 

it follows that lini ni=00 r n = 0. 

Interval of convergence. In general a convergent series, in a 
variable, x say, is convergent only for values of x in a certain 
interval, say from x = a to x=b. The series is then said to con- 
verge within the interval (a, b), and this interval is called the 
interval of convergence. 



145, 146.] INFINITE SERIES. 237 

Thus in series (2), Art. 142, the interval of convergence extends from 
x = — 1 to x = + 1. In this case, as in many others, the series is not conver- 
gent for the values of x (in this case — 1 and + 1) at the extremes of the 
interval. In some cases series are convergent for the values of the variable at 
the extremes of the interval of convergence as well as for the values between ; 
in other cases a series may be convergent for the value of the variable at one 
extreme of the interval but not for the value at the other. 

Power series. Series of the type 

a + axoc + a 2 ^ 2 + "• + a n x n •••, 

in which tlie terms are arranged in ascending integral powers of x 
and the coefficients are independent of x, are called power series 
in x. A power series may converge for all values of x, but in 
general it will converge for some values of x and diverge for others. 

Theorem. In the latter case the interval of convergence ex- 
tends from some value x = — r to the value x = + r ; i.e. the value 
x = is midway between the values of x at the extremes of the 

Divergent Convergent Divergent 

-r Q +r 

Fig. 90. 

interval of convergence. Thus in the power series (2), Art. 142, 
the interval of convergence extends from — 1 to +1. This theo- 
rem may be graphically represented, or illustrated, by Fig. 90. 
(For proof of the theorem see Osgood, Infinite Series, Art. 18.) 

146. Tests for convergence. Two simple tests for convergence 
will now be shown. For nearly all the infinite series occurring 
in elementary mathematics these tests will suffice to determine 
whether a series is convergent or divergent. These two tests are : 
(A) the comparison test and (B) the test-ratio test. 

A. The comparison test. Let there be two infinite series, 

u + u x + u 2 -\ h^n-i + ^H , (1) 

and v + i\ + v 2 + ••• + v n _j + v n + ••• (2) 

If series (1) is convergent, and if each term of series (2) is not 
greater than the corresponding term of series (1) (i.e. if v n <^ u n 
for each value of n), then series (2) is convergent. If series (1) 



238 DIFFERENTIAL CALCULUS. [Ch. XV. 

is divergent, and if each term of series (2) is greater than the 
corresponding term of series (1), then series (2) is divergent. 
Two series which are very useful for purposes of comparison are : 

(a) The geometric series 

a + ar + ar 2 -J- • ••, 

which is convergent when | r | < 1, divergent when | r | > 1. 

(6) The series 1 +-i + ^ + — + — , 

which is convergent when p > 1, divergent when p ^ 1 (see Art. 
145). 

Ex.1. The series 1 + i + tV + ei + •" 

is convergent, for it is term by term not greater than the geometric con- 
vergent series 

1 + 4 + T<5 + 6 4 + "•• 

_B. The test-ratio test. In series (6), Art. 142, the ratio 

Ull+1 (3) 

is commonly called the test-ratio. If when n increases beyond all 
bounds this ratio approaches a definite limit which is less than 1, 
then the series is convergent. For, suppose that ratio (3) is finite 
for all values of n, and suppose that after a certain finite number 
of terms, say m terms, it is less than a fixed number R which is 
less than 1. Now 

s = u x + u 2 H h u m + u m+1 + u m+2 + ■••• 

The sum of the first m terms is finite. Since 

it follows that the series beginning with u m is less than the 

u m (l + R + R* +».), 
and, accordingly, is less than 



geometric series 



1 
U ™±-R- 



14(3.] INFINITE SERIES. 239 

Hence s < s m + u m > 

and tlins the series is convergent. 

If when n increases beyond all bounds the test-ratio approaches 
a definite limit which is greater than 1, the series is divergent. 

Ex. 2. Prove the last statement. 

If the limiting value of the test-ratio is + 1 or — 1, further special investi- 
gation is necessary in order to determine whether the series is convergent or 
divergent.* 

Thus the quality of the series, as regards its convergency or 
divergency, depends upon 

lim^^ti. 

EXAMPLES. 

3. Find whether the following series are convergent or divergent : 

(1) TV^ + ^4V- (2)1 + ^fl + 4l + -' 

(5) l+l + l + i + 1.... 
W 2p Sp 4p 6p 

4. Examine the following series for convergency : 

(1) l + 3x + 5£ 2 + 7x 3 +9^ + —, (2) l 2 + 2 2 x + 3 2 a: 2 + 4 2 x 3 4-5 2 x 4 4-..., 

/>• 1"2 f& -T^ T T^ Y& or4 

( 3) l + f+|j + *+^+™. (4) iVjTTi + iJT, i + ^8 + -' 

£ , X 2 , X 3 , a* /AN „ * 3 , £ 5 z 7 . .. 



« 1 +i+f+S + - + sffT + - w *rli 



5! 7 ! 



* A series in which the absolute value of the test-ratio tends to the limit 

unity as n increases, will be absolutely convergent if, for all values of n after 

some fixed value, 

this absolute value <■ 1 ^-— , 

— n 

where c is a positive quantity independent of n. (For a proof of this general 
theorem, see Whittaker, Modem Analysis, Art. 13.) 



240 DIFFERENTIAL CALCULUS. [Ch. XV. 

147. Differentiation of infinite series term by term. It is be- 
yond the limits of a short course in Calculus to investigate the 
conditions under which an infinite series can properly be differ- 
entiated term by term ; in other words, to determine what condi- 
tions must be satisfied in order that Equation (4), Art. 143, (e), 
may be true.* 

It must suffice here merely to state the theorem that applies to 
most of the series that are ordinarily met in elementary mathe- 
matics, viz. : 

A power series f can be differentiated term by term for any value 
of x within, but not necessarily for a value at, the extremities of its 
interval of convergence. (For proof see Osgood, Infinite Series, 
Art. 41.) See Art. 197. 

148. Examples in the differentiation of series. 

In this article the results are obtained by application of the 
theorem in Art. 147. 

EXAMPLES. 

1. It is known that (see Art. 152, Ex. 7) 

e- = H-x+|l + |i+ ••-, (1) 

the second member of (1) is a power-series ; accordingly, the theorem of 
Art. 147 applies. 

On differentiation of each member of (1), 

rJ r 2 

j_ (e * )=1 +.+!_+... 

= e s , as already known. 

2. It is known that (see Art. 152, Ex. 2) 

smx=x- — + — (1). 

3! 51 ^ J 

On differentiation, cos x = 1 - — + (2) . (See Art. 152, Ex. 5. ) 

2 14! 

3. Derive expansion (1) from (2) of Ex. 2 by differentiation. 

4. When — l'<as<l, 

1 



1 -x 



1 +x + x 2 + x*+ .... (1) 



* On this, see Infinitesimal Calculus, Art. 173, especially Note 2 of that 
article for references. t See page 237. 



147, 148.] INFINITE SEEIES. 241 

On differentiation, 

= 1 +2x + 3x 2 + 4a; 3 + -. 

(1 - xy 

On differentiation and division by 2, 

= 1(1-2 + 2 .3x + 3.4x 2 + ...)• 



(1-x) 3 2 
5. Show by successive differentiation of the members of Ex. 4 (1) that 

= (1 - *)- = 1 + mx + m(m - 1 W" " = 1)( ^~ 2) ^ 3 + v 



(l-x) m v 1-2 1.2-3 

6. It is known that (see Art. 150, Ex. 2) 

log(l + x) =x-|x2 + ix 3 -, (1) 

a series which is convergent if — 1 < x <^ 1. 
On differentiation in (1), 

— L-=l-as + a*...; (2) 

1 + x 

which is true if — 1 < x < 1, but not if x = 1. 



CHAPTER XVI. 

TAYLOR'S THEOREM. 
(See N.B. at beginning of Chapter XV.) 

149. Taylor's theorem is one of the most important theorems 
in the calculus. It has a wide application, and several important 
series, for example, the binomial series (see Ex. 6, Art. 150) can 
be derived by means of it. Let f(x) be a function of x which is 
continuous throughout the interval from x = a to x = b, and which 
also has all its derivatives continuous in this interval. Now let 
x receive an increment h. Taylor's theorem is a theorem which 
gives the development of the function f(x -f- h) in a power series 
in //. The power series itself is called Taylor's series. (See Note 
2, Art. 152.) 

N.B. In reading this chapter it is better to take up Art. 154 
first. 

150. Derivation of Taylor's theorem. Let f(x) and its first n 
derivatives be continuous in the interval from x = a to x = b. It 
has been proved in the extended theorem of mean value (Art. 113, 
Eq. 4) that, on denoting 

6 — a by h, 



f(a + ft) =/(«) + hf(a) + £f"(a) + |l/'» (a) + 
n\ 



(8) 



If x and x-\-h denote any values in the interval for which f(x) 
and its first n derivatives are continuous, 

i.e. if a-^Lx^b, and a < x + h ^ b, 
242 



149,150.] TAYLOR'S THEOREM. 243 

then theorem (8) holds true for f(x + h). On replacing a in 
(8) by x there is obtained 

fix + h) = /(as) + hf'{x) +|!/H(aO + ... + 7^jy]/ n_1 0*0 

+ ^/(»>(a> + 0Jk),O<0<l. (9) 

This is Taylor's theorem with the remainder, the last term of the 
second member being denoted as the remainder. In formula (9) 
x and x + h must both be in the interval of continuity ; in any 
particular application of this formula, x has a fixed value and h 
varies. Theorem [or formula] (9) is true for all functions which, 
with their first ?i-derivatives, are continuous in the assigned inter- 
val of continuity. If all the derivatives of f(x) are continuous in 
the interval, and if 

\im n J^-f^{x + Oh) = 0, 

n I 

then A» + *)=/(*)+*/'C*)+^/' / (»)+^/ ,,, .C») + -". (10) 

For (by Art. 145) the infinite series in the second member converges 
to the value of f(x -f- k) and, accordingly, represents the function 
f(x-\-h). Formula (10) is called Taylor's theorem, and the 
series is called Taylor's series. In (9) and (10) h may be positive 
or negative, so long as x and x-\-h are in the interval of con- 
tinuity. " Hie remainder" the last term in (9), represents the 
limit of the sum of all the terms after the nth term of the infinite 
series in (10) ; it is the amount of the error that is made when 
the sum of the first n-terms of the series is taken as the value of 
the function. 

Note. The method in Art. 110 of proving the theorem of mean value was 
first given by Joseph Alfred Serret (1819-1885), professor of the Sorbonne in 
Paris, in his Cours de calcul differentiel et integral, 2 e 6"d., t. I., page 17 seq. 
The above proof of Taylor's theorem appears in Harnack's Calculus (Cath- 
cart's translation, Williams and Norgate), pages 65, 66, and in Gibson's 
Calculus, pages 390-393. The proof in Echols's Calculus (p. 82) is likewise 
based on the theorem of mean value. 

Taylor's theorem and series are important in the theory of functions of 
a complex variable, and are more fully investigated in that subject. 



244 DIFFERENTIAL CALCULUS. [Ch. XVI. 

EXAMPLES. 

» 

1. Express log (x + h) by an infinite series in ascending powers of h. 
Here f(x + h) = log (x + h). 

.\/(x) =k)gX, 
X 

/"c*o = -4 

x 2 

/»"(«) = * etc. 

a; 3 

.-. log (x + A) = log* + * - W- + _^__ J}L + .... 
a: 2x 2 3x 3 4 a;* 

Here x must not be 0, for then f(x) = - bo , and thus is discontinuous for 
x = 0. The series is evidently more rapidly convergent the smaller is h and 
the larger is x. 

On putting x = 1 and A = 1, this result gives 

log2 = l--i+i--i.+ ..., 
as found in Ex. 3, Art. 198. 

If the finite series in (9) is used, then 

log (X + h) = logX + * +^i -4- ... + (- I)"" 1 , z - ~ r , < 0< 1. 

x 2 x 2 ?i ! (x + 0/i) n 

Here, if x % h = 1, 

log2-l-i + i-i + -. + (-l)»-i 



n(l + ey 



On interchanging /i and x in formula (10), if that can be done 
in the interval of continuity, there is obtained the following 
form of Taylor's theorem : 

f(x + h) = f(h)+acf(h) +|y/"W + |^/'"W+... ? (11) 

a form which is often useful. Similarly in the case of formula (9). 

2. Express log (x + Ji) by an infinite series in ascending powers of x. 

Here/(x + A)=:log(x + A). .'. f(h) = \ogh, f'(h)=\, f"(h) = -^- etc. 

h h 1 

.-. log (x + h) = log h + - - — + — . 

Ifh = l i l g(l + x)-x-| 2 + |-^+-, 

as otherwise obtained in Ex. 3, Art. 198. 



150.] TAYLOR'S THEOREM. 245 

3. Represent sin (x + h) by an infinite series in ascending powers in h. 
Here f(x + h) = sin (x + K). .: /(x) = sin x, /'(x) = cos x, f"(x) = — sinx, 
etc. 

Hence, on using formula (10), 

sin (x + h) = sin x + h cos x sin x cos x H sin x + •••. 

2! 3! 4! 



^o of a radian (i.e. 34' 22".65). 



Then 



— -;:■:.- — cos — sin — cos — + •••. 

\S 100 J 3 100 3 (100)- 2! 3 (100)3 3! 3 

This is a rapidly convergent series. 

Now sin — = .86603, cos — = .50000. On making the computations, it will 

be found that, to Jive places of decimals, sin 60° 34' 22". 65 = .87099. 

Note. The last exercise is an example of one of the most useful practical 
applications of Taylor's theorem. Namely, if a value of a function is 
known for a particular value of the variable, then the value of the function 
for a slightly different value of the variable can be computed from the known 
value by Taylor' 's formula. (See Art. 27, Notes 1,3; Art. 82, Note 3.) 

4. Expand sin (x + h) in a series in ascending powers of x. 

In this case form (11) is to be used. Here fix + K) = sin (x + h) . 
.-. f(h) = sin h, f'(h) = cos h, f"{h) = - sin h, f"{h) = - cos h, etc. 

X 2 X 3 

.-. sin (x + h) = sin h + x cos h sin h cos h + •••. 

V J 2! 31 

On letting h = 0, the following important series is obtained : 

sinx = x - — + — . 

3! 5! 

5. Expand cos (x + K) in series, (a) in ascending powers of h, (b) in 
ascending powers of x. From the latter form deduce the series 

COSX = l -*-+*- . 

2! 4! 

6. Expand (x + h) m by Taylor's formula in a power series in h, and 
thus obtain the Binomial Expansion 

(x + h) m = x m + mx m -^h + m - m ~ 1 x m - 2 h* 4- —. 

1 • 6 

(This series is convergent for h < 1, divergent for h > 1. The case in which 
h =± 1 requires special investigation.) 



« 
246 DIFFERENTIAL CALCULUS. [Ch. XVI. 

7. Given that f(x) = 4 x 3 - 3 x 2 + 7 x + 5, develop /(x + 2) and f(x - 3) 
by Taylor's expansion. Then find J\x + 2) and /(x - 3) by the usual 
algebraic method, and thus verify the results. 

8. (1) Assuming sin 42°, compute sin 44° and sin 47° by Taylor's 
expansion. (2) Assuming cos 32°, compute cos 34° and cos 37° by Taylor's 
expansion. (3) Do further exercises like (1) and (2). 

9. Derive log(x + h) = \ogh + |_^ + j£---^ + ..., when \x\<l; 

log(x+70=loga + ^--^ + ^--..., when |x|>l. 
x 2 x 2 3 x 3 

10. Show that 

log sin (x + a) = log sin x + acotx - — csc 2 x + — C0SX + •••• 

2 3 sin 3 x 

151. Another form of Taylor's theorem. This form expresses 
f(x) as a series in ascending powers of (x — a). On writing x for 
b in Art. 113, Eq. (3), and in the value of x n , two lines after that 
equation, there is obtained 

z n— 1 ! 

+ (ag ~f )n /* n) [CT+e(a;-CT)],o<e<i. (l) 

nl 

If all the derivatives of f(x) are continuous in the assigned 
interval, and 

lim Mi00 (^ ~ a )> »)[a + 0(* - a)] = 0, 

?i ! 

then (Art. 145) the infinite series /(a) + (a? — a)/'(a)-f-J(#— a) 2 f"(a) 
+ ••• represents the function /(a?) * ; i.e. i 

/Cos) = /(«)'+ (op - «)/'(ffl)l (ag ~ f g) V («)+ (a? 3 ! CT)8 / /,, W+ - 

+ (g-«)* /( n) («0+'". (2) 

Forms (1) and (2) for Taylor's theorem and series, are fre- 
quently useful. The last term in the finite series (1) is Lagrange's 
form of the remainder in Taylor's series. (See Note 4, Art. 152.) 



* Except in some rare cases. 



150, 152.] TAYLORS THEOREM. 247 

EXAMPLES. 

1. Express 5 x 2 + 7 x + 3 in powers of x — 2. 

Here f(x) = 5 x 2 + 7 a; + 3, .-. /(2) - 37, 

/'(as) = 10 as + 7, /'(2)=27, 

/»(x)=sl0, /"(2) =10, 

/'"(a;)=0, /'"(2)=0. 

Now by (2), /(x) =/(2) + (x - 2)/' (2) + (x ~ 2) > (2) + ;... 



.-. 5 x 2 + 7 x +3 = 37 + 27(x - 2) + 5(x - 2) 



2. Express 4 x 3 — 17 x 2 + 11 x + 2 in powers of x + 3, in powers of 
x — 5, and in powers of x — 4, and verify the results. 

3. Express 5 y± + 6 y 3 — 17 y 2 + 18 y — 20 in powers of ?/ — 4 and in 
powers of y + 4, and verify the results. 

Note. Exs. 1-3 can be solved, perhaps more rapidly, by Horner's process. 
(See text-books on algebra, e.g. Hall and Knight's Algebra, § 549, 4th edition, 
1889.) 

4. Develop e x in powers of x — 1. 

5. Show that -= - - — (x _ a ) + — (x - a) 2 - — (x - a) 3 + • •-, when x 

x a a 2 a 3 a 4 

varies from x = to x = 2 a. 

6. Show that log x = (x - 1) - \(x - l) 2 + i (x - l) 3 is true for 

values of x between and 2. 



152. Maclaurin's theorem and series. This is a theorem for 
expanding a function in a power series in x. As will be seen 
presently, it is really a special case of Taylor's theorem. 

Let f(x) and its first n derivatives be finite for x = and be 
continuous for values of x in the neighborhood of x = 0. 

In form (9), Art. 150, put x = ; then 



/(ft) =/(0) + A/'(0) + *>(») + • • • + _^/<»-"(0) + £>>(«). 
^ ! (w — 1) ! 71 I 

On writing x for ft, this becomes 

f(x)=f(0)+xf'(0)+ ff"(0)+.-+ J^f'-»(p) + ?S<'\6x). (1) 
Z ! (n— 1)! w! 



248 DIFFERENTIAL CALCULUS. [Ch. XVI. 

If f(x) and all its derivatives are finite for x = 0, and if 



x 



lim,^ —fW$(x) = 0, then 



n 



OC 2 *,, A x , , 0!». 



/[*) =/(0) + a>/'(0) +|j/"(0) +...+^n) (0 ) + .... (2) 

This is known as Maclaurin's theorem, and the series is called 
Maclaurin's series. The last term in (1) is called the remainder in 
Maclaurin's series. It is the limit of the sum of the terms of the 
series after the wth term. 

EXAMPLES. 

1. Show that formula (2) comes from form (11), Art. 150, on putting 
h = ; show that this has practically been done in the derivation above. 
Show that formula (2) comes from form (2), Art. 151, on putting a = 0. 

2. Develop sin £ in a power series in x. 

Here f(x) = sin x. :. /(0) = 0, 

.\/'(x)=COBJB, /'(0) = 1, 

/"(£)=- sin x, /"(0) = 0, 

f"(x) = - cos x, /'"(0) = - 1, 

/ iv (x) = sinx, /iv(0) = 0, 
etc. etc. 

' ■•■—_=- S + f!-n + - + S^^- 1+ - (A) 

(Compare Ex. 2 above and Ex. 4, Art. 150.) 

On applying the method of Art. 146 it will be found that the interval of 
convergence is from —no to + go. 

3. Calculate sin ( r L radian), i.e. sin 5° 43' 46".5. 

By A, sin (.1 radian) = .1 - -^^ + ^^ = .09983. 

4. Calculate sin (.5 r ) and sin (.2 r ) to 5 places of decimals. (For results, 
see Trigonometric Tables.) 

5. Showthat C0S x = l-^ + ^-^+.», (B) 

11 4 ! o ! 

and show that the interval of convergence is from — oo to + go . 

6. To 4 places of decimals calculate the following: sin(.3 r ), cos(.2) r , 
sin (.4 r ), cos (.4'"). (See values in Trigonometric Tables.) 



152.] TAYLOR'S THEOREM. 249 

7. Show that e* = 1 + 05 + |^ + |^+ — , (C) 

and show that this series is convergent for every finite value of x. 

8. Substitute 1 for x in C, and thus deduce 2.71828 as an approximate 
value of e. 

9. Assuming A and B deduce that the sine of the angle of magnitude zero, 
is zero, and that the cosine of this angle is unity. 

Note 1. Expansions A and B were first given by Newton in 1669. He 
also first established series C. These expansions can also be obtained by the 
ordinary methods of algebra, without the aid of the calculus. For this 
derivation see Chrystal, Algebra, Part II., Chap. XXIX., § 14, Chap. 
XXVIII., § 5, and the texts of Colenso, Hobson, Locke, Loney, and others, 
on what is frequently termed Analytical Trigonometry, or Higher Trigo- 
nometry. [This subject is rather to be regarded as a part of algebra 
(Chrystal, Algebra, Part II., p. vii).] Also see article "Trigonometry" 
(Ency. Brit., 9th ed.). 

10. Develop the following functions in ascending powers in x : (1) sec x ; 
(2) log sec x; (3) log (1 + x), tan- 1 a;, sin" 1 a: (see Art. 198, Exs. 1, 2, 3.) 

11. Show that tan x = x + $ x 3 + T 2 5 xb + imr x7 + "• 
By this series compute tan (.5 r ), tan 15°, tan 25°. 

12. Find: (1) (e'cosxdx: (2) C-dx: (3) \~ e'** dx. 



(1) (VcosxcZx; (2) C— dx\ (3) (*< 



Note 1 a. The integral in Ex. 12 (3) is important in the theory of probabili- 
ties. If the end-value x is qo, the value of the integral is \Vtt. (Williamson, 
Integral Calculus, Ex. 4, Art. 116.) 

13. Assuming the series for sin x, prove Huyhen's rule for calculating 
approximately the length of a circular arc, viz. : From eight times the chord 
of half the arc subtract the chord of the whole arc, and divide the result by 
three. 

14. State Maclaurin's theorem, and from the expansion for tana; find 
the value of tan x to three places of decimals when x = 10°. 

15. Show that cos* x = 1 - — x 2 + n & n ~ 2>> x* . 

2 ! 4 ! 

Note 2. Historical. Taylor's theorem, or formula, was discovered by 
Dr. Brook Taylor (1685-1731), an English jurist, and published in his Metho- 
dus Incrementorum in 1715. It was given as a corollary from a theorem in 
Finite Differences, and appeared without qualifications, there being no refer- 
ence to a remainder. The formula remained almost unnoticed until Lagrange 
(1736-1813) discovered its great value, investigated it, and found for the 



250 DIFFERENTIAL CALCULUS. [Ch. XVI, 

remainder the expression called by his name. His investigation was pub- 
lished in the Memoires cle VAcademie de Sciences a Berlin in 1772. "Since 
then it has been regarded as the most important formula in the calculus." 

Maclaurin's formula was named after Colin Maclaurin (1698-1746), pro- 
fessor of mathematics at Aberdeen 1718 ?-1725, and at Edinburgh, 1725-1745, 
who published it in his Treatise on Fluxions in 1742. It should rather be 
called Stirling' 1 s theorem, after James Stirling (1690-1772), who first an- 
nounced it in 1717 and published it in his Methodus Differential in 1730. 
Maclaurin recognized it as a special case of Taylor's theorem, and stated 
that it was known to Stirling ; Stirling also credits it to Taylor. 

Note 3. Taylor's and Maclaurin's theorems are virtually identical. It 
has been shown in Art. 152 that Maclaurin's formula can be deduced from 
Taylor's. On the other hand, Taylor's formula can be deduced from Mac- 
laurin's ; e.g. see Lamb's Calculus, page 567, and Edwards's Treatise on 
Differential Calculus, page 81. 

Note 4. Forms of the remainder for Taylor's series (2), Art. (151). 
Lagrange's form of the remainder has already been noticed in Art. 151. 
Another form, viz. 

( *~(w "-l") f )W ~V (w) [« + *(* ~ ")]> <^<!> 

was found by Cauchy (1789-1857), and first published in his Lecons sur le 
Calcnl infinitesimal in 1826. A more general form of the remainder is the 
Schlomilch-Boche form, devised subsequently, viz. 

(x n) . (1 ey-, + _ 

(n — Y)\p 

This includes the forms of Lagrange and Cauchy ; for these forms are ob- 
tained on substituting n and 1 respectively for p. (The 0's in these forms 
are not the same, but are alike in being numbers between and 1.) In par- 
ticular expansions some one of these forms may be better than the others for 
investigating the series after the first n terms. 

Note 5. Extension of Taylor's theorem to functions of two or more 
variables. For discussions on this topic see McMahon and Snyder's Calcu- 
lus, Art. 103 ; Lamb's Calculus, Art. 211 ; Gibson's Calculus, § 157. 

Note 6. Keferences for collateral reading on Taylor's theorem. 
Lamb, Calculus, Chap. XIV. ; McMahon and Snyder, Diff. Cal., Chap. IV. ; 
Gibson, Calculus, Chaps. XVIII., XIX. ; Echols, Calculus, Chap. VI. 

153. Relations between trigonometric (or circular) functions and expo- 
nential functions. The following important relations, which are extremely 
useful and frequently applied, can be deduced from the expansions for sin x, 
152. 



152, 153.] TAYLOR'S THEOREM. 251 

The substitution of ix for x in C gives 

e ix = 1 - |i + £ - -. + i(x - 1^ j + £ - ...\ = cos a? + i sin a?. (1) 
The substitution of — ix for x in C gives 
<T te = l - j£ + £ *(» _ |! + ^ - ...A = cos as - i sin a% (2) 

4 ! 4 ! ^ o ! o I / 

From (1) and (2), on addition and subtraction, 

cos x = e \ e (3), sinic = ^ -? (4) 

On putting 7r for x in (1), there is obtained the striking relation 

e iir = -l. (See Art. 38, Note on e.) 

Note 1. The remarkable relations (l)-(4), by which the sine and cosine 
of an angle can be expressed in terms of certain exponential functions of the 
angle (measured in radians), and conversely, were first given by Euler 
(1707-1783). (In connection with the expansions in Arts. 152, 153, see the 
historical sketch in Murray's Plane Trigonometry, Appendix, Note A ; in 
particular pp. 168, 169.) 

Note 2. Results (l)-(4) can also be deduced by the methods of ordinary 
algebra; see Note 1, Art. 152, the references therein, and Chrystal's Algebra, 
Part II., Chap. XXIX., § 23. 

EXAMPLES. 

1. From (3) and (4) deduce that cos 2 x + sin 2 x = 1. 

_ _, . pix o-ix 

2. Show that tan x = — . 



6 IX _|_ Q-IX 

3. Express cot x, sec x, cosec x, in terms of exponential functions of x. 

Note 3. Since, by (1), e<* = cos <j> + i sin <f>, and e int > = cos n <p + i sin n 0, 
and since (e**) n = e in >, it is evident that 

(cos <{> + i sin 40 w = cos n$ + i sin n$, 

for all values of n, positive or negative, integral or fractional. 

This very important theorem is called Be Moivre's theorem, after its dis- 
coverer Abraham de Moivre (1667-1754), a French mathematician who 
settled in England. It first appeared in his Miscellanea Analytica (London, 
1730), a work in which "he created 'imaginary trigonometry.' " [On Be 
Moivre' s theorem, and results (l)-(4), see Murray, Plane Trigonometry, 
Art. 98, and Appendix, Note D ; and other text-books on Trigonometry.] 

X.B. The article on Hyperbolic Functions, Appendix, Note A, may be 

conveniently read at this time. 



252 DIFFERENTIAL CALCULUS. [Ch. XVi. 

154. Another method of deriving Taylor's and Maclaurin's series. 

Following is a method which is more generally employed than 
that in Arts. 150 and 152 for finding the forms of the series of 
Taylor and Maclaurin. 

A. Maclaurin's series. Let f(x) and its derivatives be con- 
tinuous in the neighbourhood of x = 0, say from x = — a to x = a. 
Suppose that f(x) can be expressed in a power series in x conver- 
gent in the interval —a to + a. That is, assume that (for 
— a < x < a) there can be an identically true equation of the 

f0rm /(a?) = A + A x x + A 2 x> + -Af* + ... + A& + -. (1) 

The coefficients A , A x , A 2 , •••, ^ n , •••, will now be found. It 
has been seen in Art. 147 that if Equation (1) is identically true, 
then the equation obtained by differentiating both members of (1), 

Vlz - f(x) = A x + 2 A 2 x + 3 Ax 2 + • • • + nA n x n ~ x + ■ ■ ., 

also is identically true for values of x in some interval that 
includes zero. For the same reason the following equations, 
obtained by successive differentiation, are also identical in inter- 
vals that include zero, viz. : 

f"(x) = 2 A 2 + 2 • 3 A 3 x + ... + n(n - 1) A n x n ~ 2 + —, 

f'"(x) = 2 • 3 • A 3 + ... + n(n - 1) (n - 2) A^" 3 + •», 



/W(a;) = 7i.w-1 • rc-2 2 ■H + ..., 

On putting # = in each of these identities it is found that 

Hence, on substitution in (1), 

/(x)=/(0)+a;/-'(0) + |:/"(0) + g/'»(0)+...+^/<»'(0)+-, (2) 

which is Maclaurin's series (Art. 152). 

B. Taylor's series. Let f(x) and its derivatives be continuous 
in the neighbourhood of x = a, say from x = a — h to x = a -f- h. 
Suppose that f(x) can be expressed in a power series in x — a 



154.] TAYLOR'S THEOREM. 253 

which, is convergent in the neighbourhood of x = a. In other 
words, suppose that there is an identically true equation of the 
form 

f(x) = A + A,(x -a) + A 2 (x- a) 2 + A 3 (x - a) 3 + • -. 

+ A(* -«)" + -• (3) 

Then, as in case A, the following equations, which are obtained 
by successive differentiation, also are identically true for values 
of x near x = a, viz. : 

f'(x)=A l + 2A 2 (x-a) + 3A 3 (x-ay+'.' + nA n (x-a) n - 1 +>.', 

f"(x) = 2A 2 +2 .3A 3 (x-a) + --.+n. n-L A n (x- a) B - 2 +..., 

f"(x)=2 • 3 • A 3 + -. + n • n - 1 • n- 2 . J.(»- «) n " 3 + -, 

/*(a>) = %.%-! • w-2. ...2- 1 • A + — , 

On putting x = a in each of these identities it is found that 
A=f(a), A^fXa), A 2 = £^fl, A = ^., .-, 



A,= 



f in) (a) 



Hence, on substitution in (3), 



2! 
+ ^f £ / w («) + -, (4) 

which is series (2), Art. 151. 

If in (4) x is changed into x-\-a, then 

f(x + a) =f(a) + xfXa)+ff'Xa)+... + ^f«Ka) + ■", (5) 

which is series (11), Art. 150, with a written for h. On inter- 
changing a and x in (5), form (10), Art. 150, is obtained. 

Note. On the proof of Taylor's theorem. The above merely shows the 
derivation of the form of Taylor's series. It is still necessary to examine into 
the convergency or divergency of the series and to determine the remainder 



254 DIFFERENTIAL CALCULUS. [Ch. XVI. 

after any number of terms. The investigation of the validity of the series is 
a very important matter in the calculus. For this investigation see, among 
other works, Todhunter, Biff. Cal., Chap. VI. ; Williamson, Diff. Cal., 
Arts. 73-77 ; Edwards, Treatise on Diff. Cal., Arts. 130-142 ; McMahon 
and Snyder, Diff. Cal., Chap. IV. ; Lamb, Calculus, Arts. 203, 204; article, 
"Infinitesimal Calculus*' (Ency. Brit., 9th ed., §§ 46-52). 

155. Application of Taylor's theorem to the determination of con- 
ditions for maxima and minima. This article is supplementary to 
Art. 76. Let f(x) be a function of x such that f(a + h) and f(a — h) 
can be developed in Taylor's series ; and let it be required to 
determine whether /(a) is a maximum or minimum value of f(x). 
On developing f(a — h) and /(a + h) by formula (9), Art. 150, 

f(a - h) =/(«) - hf(a) + |-/»(a) - ~f"(a) + ... 



+ LJ£r f e»(a-eji), . (i) 



f(a + h) =f(a) + hf'(a) + £/»' (a) + |! f" (a) + ... 

+ J £f n) (a + eji), (2) 

in which 2 and 2 lie between and 1. 

Suppose that the first n — 1 derivatives of f(x) are zero when 
x = a, and that the nth derivative does not vanish for x = a. Then 

f(a - h) -/(o) = £=-^>>(a - eji), (3) 






f(a + h) -/(a) = ^«>(a + W- W 

It follows from the hypothesis concerning /(or) that the signs of 
f- n \a — 0Ji) and jf (n) (a + 6 2 h), for infinitesimal values of h, are the 
same as the sign of f (r, \a). From (3), (4), and the definitions of 
maxima and minima, it is obvious that : 

(a) Ifn is odd, the first members of (3) and (4) have opposite 
signs, and consequently, f(a) is neither a maximum nor a minimum 
value off(x); 

(b) If n is even and f (n) (a) is positive, the first members of (3) 
and (4) are both positive, and consequently, f(a) is a minimum 
value off(x) ; 



155, 156.] TAYLOR'S THEOREM. 255 

(c) If n is even andf (n) (a) is negative, the first members of (3) 
and (4) are both negative, and consequently, f(a) is a maximum 
value off(x). 

The condition for maxima and minima that was deduced in 
.Art. 76, (c), is a special case of this, viz. the case in which n = 2. 

156. Application of Taylor's theorem to the deduction of a theorem 
on contact of curves. This article is supplementary to Art. 95. 
(See Art. 95, Note 4.) 

Theorem. If two curves have contact of an even order, they cross 
each other at the point of contact; if two curves have contact of an 
odd order, they do not cross each other at the point of contact. 

Let the two curves y = <f>(x) and y = if/(x) (1) 

have contact of the nth order at x = a. Then 

4(a) = tfa)> +'(«) = f H +"(«) = <A», " •, </> (n) («) = «A (n) (4 ( 2 ) 

Now compare the ordinates of these curves at x = a — h, i.e. com- 
pare <f>(a — h) and if/(a — h); also compare the ordinates at x = a + h, 
i.e. compare <f>(a + h) and if/(a -f- h). Let it be further premised 
that <f>(a ± h) and if/(a ± h) can be expanded in Taylor's series. On 
using Taylor's theorem (form 9, Art. 150), and remembering 
hypothesis (2), it will be found that 

4(a -K)- fa - h) = t^ [^u( fl - ejt) - ^"+«(a - W)l (3) 
tia + ft) - itfa + *) = ,-^r-. [<P +1) (a - ftft) - ^ n+l) (a - OJCj], (4) 

V* T L ) - 

in which the four 0's all lie between and 1. 

Let h approach zero; then, by the premise above, the signs 
of the expressions in brackets are the same as the signs of 
[cf> <n+1) (a) — i//' l+1) (a)]. Hence, if n is odd, the first members of (3) 
and (4) have the same sign, and, accordingly, the curves do not 
cross; if n is even, these first members have opposite signs, and, 
accordingly, the curves do cross. 

Ex. Accompany the proof of this theorem with illustrative figures. 



256 DIFFERENTIAL CALCULUS. [Cii. XVI. 

157. Applications of Taylor's theorem in elementary algebra. Let 
fix) be a rational integral function of x, of the nth. degree say. 
Then f {n+1 \x) and the following derivatives are all zero. Hence, 
Taylor's series for f(x + h) in ascending powers of either h or x 
[see forms (10) and (11), Art! 150] is finite. That is, 

f(x + h) =f(x) + hf(x) + |/»(*) + - + ^/""(*), (1) 

f(x + K) =f(h) + xfQi) + g/»(ft) + - + ^/<"»(/0- (2) 



n\ 



A rational integral function f(x) of the nth degree can also be 
expressed in a finite series in ascending powers of a: — a [see 
form (2), Art. 151]. That is, 

/(x)=/(fl) + (x _ a)/(a) + (^> W+ ... + (^I>5 (a) . (3) 

Exercise. See Ex. 7, Art. 150, and Exs. 1, 2, 3, Art. 151. 

Note 1. Let f(x) be as specified above. In general the calculation of 
f(x + h) and the expression of f(x) in terms of x — a, can be more speedily 
effected by Homer's process.* This process is shown in various texts on 
algebra; e.g. Hall and Knight's Algebra (4th edition), Arts. 549, 572. 

Note 2. Eor an application of Taylor's theorem to interpolation, 

see McMahon and Snyder, Calculus, Note, pp. 325, 326. 

Note 3. In expansion (10), Art. 150, if h is a differential dx of x, then 
h, A 2 , h s , •••, are respectively differentials of x of the first, second, thi>d, ••-, 
orders; and hf(x), h 2 f"(x), h s f'"(x), •••, are respectively differentials of 
f{x) of the first, second, third, •••, orders. If h (or dx) is an infinitesimal, 
these differentials are also infinitesimals of the respective orders mentioned. 

* William George Horner (1786-1837), an English mathematician, who 
discovered a very important method of finding approximate solutions of 
numerical equations of any degree. 



CHAPTER XVII. 



APPLICATIONS TO SURFACES AND TWISTED CURVES. 

158. Introductory. 

(a) Plane curves of one parameter. In the case of a circle 



x 2 -\-y 2 



(1) 



(2) 



the varying positions of a point (x, y) on the 
circle may be described by giving values to 
6 in the equations 

x = a cos 6, 
y = a sin 0. 

Here denotes the angle made with the 
x-axis by the radius drawn from the centre 
to the point. 

In the case of the ellipse 

7^ 2 




Fig. 91. 



£ + £=1, 



(3) 



w 



or b 2 

the varying positions of a point (x, y) may be described by givin 
values to <£ in the equations 

x = a cos <j>, 1 * 

y = b sin $. J 
The equations of the cycloid, 

x = a(6 — sin0), 1 ,-v 

y = a (1 — cos 0), J 
have been used in several preceding articles. 

Variable numbers such as 0, <f>, 0, used in equations (2), (4), 
(5), are called parameters. Curves, such as the above, in whose 
equations only one parameter appears, are called curves of one 
parameter. 

* See text-books on analytic geometry. 
257 



258 DIFFERENTIAL CALCULUS. [Ch. XVII. 

(b) Twisted curves or skew curves. A twisted curve, also called 
a skew curve, is a curve which, does not lie in a plane. Thus the 
curve which is drawn on the surface of a right circular cylinder 
crossing the elements of the cylinder at any constant angle not a 
right angle, is a skew curve. 

Skew curves sometimes may be expressed in terms of one param- 
eter. Thus the equations of the curve just described, a helix, are 

x = acosO, y=asin$, z = b$. 

Here a is the radius of the cylinder, at any point is the angle 
which the projection of the radius vector of the point makes with 
the a?-axis on the xy- plane, and b is a constant depending on a and 
the constant angle at which the curve crosses the elements of the 
cylinder. (See Fig. 150, Note C. Here b = a tan a.) 

Another example of equations of a skew curve of one parame- 
ter is 

x = 2 a cos t, y = 2 a sin t, z = ct 2 . 

Tangent to a skeiv curve. A method of finding the direction of 
the tangent to a plane curve y = f(x) at any point has been shown 
in Arts. 24, 59. The method was founded on the definition that a 
tangent at any point of the curve is the limiting position of a se- 
cant drawn through that point when a neighboring point of inter- 
section of the secant with the curve approaches the first point. 
A like definition will be used in finding the direction of the tan- 
gent to a skew curve. 

(c) Direction cosines of a line. Let the line OP (or any parallel 
line US) make, angles a, (3, y, with the 
axes OX, OY, OZ, respectively. Then 

cos a, cos fi, cos y 

are called the direction cosines of the 
line. 

The direction of a line is known 
when two of them are given ; since, as 
shown in analytic geometry, 

Fig. 92. COS 2 a + COS 2 j3 + COS 2 y = 1. 




158, 159.] SURFACES AND TWISTED CURVES. 



259 



(d) It is shown in analytic geometry that if a, b, c are propor- 
tional to the direction cosines of a line ; that is, if 

a : b : c = cos a : cos /3 : cos y, 

then the values of the direction cosines are respectively, 

a b c 



Va 2 + 6 2 + c 2 Va s + & s + c s ^/tf + b' + c 2 



159. Tangent line to a twisted curve, 
curve be 



Let the equations of the 




A y, z x + a z) 



Take any point 
P on the curve ; let 
its coordinates be 
O&d yu %)• Through 
P draw any secant 
meeting the curve 
in Q. Denote the 
coordinates of Q as FlG - 93 - 

(xj + Ax, y 1 + Ay, z x + Az) . Denote the value of t at P as t x , and 
the value of t at Q as ^ -f- At. Thus Ax, Ay, Az, At are the corre- 
sponding differences between the coordinates and the parameter t 
respectively, at P and Q. 

The direction cosines of the secant PQ are proportional to 

Ax, Ay, Az ; * 
Ax Ay Az 



and hence proportional to — , — " 



At At At 



(2) 



Now suppose the secant PQ turns about P, Q moving along the 
curve until it comes to P. TJie limiting position of PQ ichen Q 
thus arrives at P is the tangent line to the curve at P. When Q ap- 
proaches P, At approaches zero, and the quantities (2) approach 

* It is shown in analytic geometry that the direction cosines of the line 
passing through the points (xi, yi, z{), (x 2 , y 2 i z 2 ), are proportional to x 2 — Xi, 
V2 — 2/1, z 2 - zi, respectively. 



260 DIFFERENTIAL CALCULUS. [Ch. XVII. 

the values — , — , — • Accordingly, the direction cosines of the 
dt' dt dt &J ' 

tangent to the curve at a point P(x 1} y 1} z-^) are proportional to the 

t o dx dy dz , , N 

values oi — , -^ — at (ah, Vu %)• 
eft' dt' dt v 1? ^ ; 

These values may be denoted by -^, -&, -^. 

J J dt dt dt 

It is shown in analytic geometry that the equations of a line 
passing through the point (x 1} y lt %) and having the direction 
cosines proportional to I, m, n, are 

x x i _ y V\ _ % z \ _ /o\ 

Z m n 

The equations of the tangent line drawn to the curve at (x 1 , y 1: Zj) 
are accordingly ^__^ ^ ^_^ ^ z _^ 

C?X! CZ?/! ffej ' 

dt dt dt 

160. Equations of a plane normal to a skew curve of one param- 
eter. A plane is said to be normal to a skew curve at a point 
when it is normal to the tangent line to the curve at that point. 

It is shown in analytic geometry that if the direction cosines 

of a line are proportional to I, m, n, the equation of the plane 

which passes through a point (x 1 , y ly z£) and is at right angles to 

that line, is _ > ., . 

l(x-xl) + m(y- 2/j) + n (z - %) = 0. (1) 

Hence, from this property, the preceding definition, and equa- 
tions (4), Art. 159, the equation of the plane which is normal to 
the skew curve (1), Art. 159, at the point (x ly y v z^) is , 

EXAMPLES. 

1. Find the equations of the tangent line and the equation of the normal 
plane which are drawn to the curve 

x = 2 a cos t, y = 2 a sin t, z = ct 2 : 

(1) at any point (xi, y\, z{) ; (2) at the point for which t = — ; (3) at the 

2 
point for which t = ir. 



159, 160.] SUB FACES AND TWISTED CURVES. 261 

(1) Here. ||=-2asiiU = - Vl , 

^ = 2 a cos t = aci, 

dt 

^ = 2ct = 2Vcz^. 
dt 



2 ax + ttc ( z - ^ ) = 0. 



(«) 



Hence the equations of the tangent line at (xi, yi, z{) are 
x - xi _ y — ?/i _ z - gi _ 

- yi «i 2 Veil 

The equation of the normal plane at {x x , y\,z{) is 

- y x ix - xi) + Xi (y - ?/i) + 2 VcilOs - si) = 0. 

This reduces to 

xiy - ijix + 2a cz x (x - z{) = 0. (6) 

(2) When £ = - . the point («i, ?/i. Si) is [ 0, 2 a, — ^ • 

Equations (a) then have the form 

_ irfc 

a? 2/ — 2 a 1 



- 2a 7T 

whence ircx + 2 a: - ^-^ = and y =2 a. 

2 y 

Equation (b) then is 

(3) When t = tt. the point (aci, ?/i, Si) is (— 2 a, 0, tt 2 c). 

The equations of the tangent line are x + 2 a = 0, 7rc?/ + as = 7r%c. 

The equation of the normal plane is 2 a?/ = 7rc (2 — ir' 2 c). 

2. Find the equations of the tangent and the equation of the normal 
plane to the helix x = a cos d, y = a sin 6, z — bd: 

(a) at any point (sbi, y\. Z\)\ (6) when 9 = 2 rr. 

Ans. (a) x ~ ' Tl = ^ ~ -' /t = z ~ * x , equations of tangent line ; 
- V\ &i b 

— yi(x — Xi) + Xi(?/ — yi) + 6(2 — ^i) = 0, equation of normal plane. 

(6) x = a. by — az — 2 abir, equations of tangent line ; 

ay + bz — 2 b 2 -rr = 0, equation of normal plane. 

(See Granville, Calculus, p. 272, Ex. 1. ) 



262 



DIFFERENTIAL CALCULUS. 



[Ch. XVII. 



161. Tangent lines to a surface at any point. Tangent plane to 
a surface at any point. Suppose a straight line is drawn through 
a point on a surface and any neighboring point, and that the 
latter point moves towards the first point along the surface. 
The limiting position of the line as the moving point approaches 
the fixed point is said to be a tangent line to the surface at this 
point.* A neighboring point may be chosen in an unlimited 
number of ways, and moreover it can approach the fixed point 
by any one of an unlimited number of paths on the surface. It 
is evident, accordingly, that through any ordinary point on a sur- 
face an unlimited number of tangent lines can be drawn. 

Theorem. All the tangent lines that may be drawn through an 
ordinary (i.e. a non-singular) point on a surface lie in a plane. 

Let the equation of the surface be 

F(x,y,z)=0. (1) 

Suppose that 

x=f(t),y = cf>(t),z = t(t), (2) 

are the equations of a curve C drawn on 
the surface through a point P(x 1} y 1} z x ). 
Then at P, the total ^-derivative of 
Fix, y, z), by (1), must be zero; that 
is, from (1) and (2), 




Fig. 94. 



dF dx dF dy dF dz 

— I • — H — 

dx dt By dt dz dt 



0. 



For P(x h y lf %) equation (3) may be written 



dF dx, dF dyi dF dz x 
dx 1 '~dt^~dy 1 '~dt^"dz 1 '~di 



= 0: 



(3) 



W 



in which — denotes the value of ■ — when x l7 y lf z 1 are substi- 
dx x dx 

d i* fj / y* 

tnted for x, y, z, and — x denotes the value of — at P. 
dt dt 



* This definition of a tangent line to a surface applies only to ordinary 
points on the surface. "Singular points" on a surface are not discussed 
here. 



161.] SURFACES AND TWISTED CURVES. 263 

According to the definitions in Arts. 159, 161, the tangent line, 
T say, drawn to the curve C at P must be a tangent line to the 
surface. By Art. 159 the direction cosines of the tangent line 
to the curve C at P are proportional to 

dx! dy x dz± ^ 

clt ' clt' dt K ' 

Equation (4) shows # therefore that the tangent line T is per- 
pendicular to a line through P, N~ say, whose direction cosines are 
proportional to QF dF dF 

"Z~> ^~» T" ( 6 ) 

ai\ dy 1 dz 1 

But T is any tangent line through P; accordingly the line JV 
is perpendicular to all the tangent lines through P. There- 
fore, all these lines lie in a plane, viz. the plane passing through 
P at right angles to JSf. This plane is called the tangent plane at P. 

The line JV, from fact (6), is perpendicular to the plane 
through P(x 1 , y 1} z{) whose equation is f 

(x - xj) — -f {y - yi ) — + (z - Zj) — = 0; (7) 

dxj dy 1 dz 1 

this, accordingly, is the equation of the tangent plane at P. 

Should the equation of the surface be in the form 

z=My), (8) 

this can be put in form (1), viz. : 

f(x,y)-z = 0. (9) 

m, dF dF dF 

Then — , — , — , 

dx r dy x dz 1 

are respectively -J-, -J- , —1, 

dxy dy x 

* It is shown in analytic geometry that if two lines are perpendicular 
to one another and their direction cosines are proportional to I, m, n, and 
?i, wi, Hi, respectively, then 

Hi + mmi + nji\ = 0. 

f By analytic geometry, the equation of a plane through a point (xi, yu &i) 
at right angles to a line whose direction cosines are proportional to Z, wi, n, is 
l(x - x x ) + m(y - ?/i) + n(z - z x ) = 0. 



26 1 DIFFERENTIAL CALCULUS. [Ch. XVII. 

Bx 1 dy l 
and (7), the equation of the tangent plane at (x r , y lf z^) becomes 

(x-x,)p+ (y - a) p - (« - *,) = 0. (10) 

ox 1 oy 1 

Note. For another derivation of (10) see Osgood, Calculus, pp. 288, 289. 

162. Normal line to a surface at any point. A line which is 
drawn through a point on a surface at right angles to the tangent 
plane passing through the point is said to be a normal to the surface. 

It has been seen in Art. 161 that the line N, which is drawn 
through the point P(x Jf y 1} Zj) and whose direction cosines are 

proportional to — , — , — , is at right angles to the tangent 
dXi dy x dz x 

plane at P. Accordingly, A 7 " is a normal to the surface at P. 

Its equations, since it passes through that point with those direc- 

tion cosines, are ^_^ ^ ^_^ % _^ 

dF ~ dF dF ' (1) 

dx 1 dy x Bz 1 

Otherwise : Since the normal at P is perpendicular to the tan- 
gent plane at P, whose equation is (7), Art. 161, the equations of 
the normal are (1).* 

When the equation of the surface has the form 

«=/(«> y), 

the equations of the normal at (x l} y lt %) [see Art. 161, (8)-(10)] are 

x — x x y — y 1 z — z x 

(2) 



dx 1 

X — X-, 

s 


tyi 

y - yi 


-1 

z — z x 


ClZy 

dx x 


dz 1 

dyi 


' -1 



These are the same as _ — = - - 



* By analytic geometry the equations of the line drawn through a point 
(xi, yi, Z\) at right angles to a plane Ix + my + nz -\- p = 0, are 
x — xi _ y — yx _ z -z x 
I m n 



162.] SURFACES AND TWISTED CURVES. 265 



EXAMPLES. 

1. Find the equation of the tangent plane and the equations of the 
normal line to the ellipsoid 

x 2 + 2 y 2 + \z 2 = 26, 
at the point (2, 3, 1). 

Here ^=2x, ^ = iy, ^ = 8*. 

dx dy dz 

At (2, 3, 1) these values are 

d^i dyi dzi 

The equation of the tangent plane, by substitution in (7), Art. 161, is 
(x - 2)4 + 0/ - 3)12 + ( 2 _l)8=0, 
i.e. 4x + 12y + 8^ = 52. 

The equations of the normal line, by substitution in (1), Art. 162, are 

x-2 _ y-3 _ g- 1 

4 12 8 ' 

which simplify to 3 x = y + 3, 2 ?/ = 3 z + 3. 

2. Find the equation of the tangent plane and the equations of the normal 
line to each of the following surfaces : 

(a) the sphere x 2 + y' 2 + z 2 + 8 x - 6 y + 4 z = 17 at the point (2, 4, 1) ; 

(6) the hyperboloid of one sheet 2 £' 2 + 3 y 2 - 7 2 2 = 38 

at the point (-3,4,2); 

(c) the hyperboloid of two sheets x 2 — 4 y 2 — 3 2 2 + 12 = 

at the point (8, - 4, 2) 

(d) the elliptic paraboloid z = x 2 + 3 y 2 at the point (2, — 3, 31) 

(e) the sphere x 2 + y 2 -\- z 2 — 12x — ±y — 6z = at the origin 
(/) the surface x 2 ■+ y 2 -iz 2 = 16 at the point (8, 4, 4). 

3. Show that the sum of the squares of the intercepts on the axes made 
by any tangent plane to the surface 

2. 2 2. 2 

x 3 + y 3 +z 3 = a 3 , 
is constant. 

4. Show that the volume of the tetrahedron formed by the coordinate 
planes and any tangent plane to the surface 



xyz = a 



is constant. 



266 



BIFFEBENTIAL CALCUL US. 



[Ch. XVII. 



163. Equations of the tangent line and the normal plane to a 
skew curve. * 

A curve may be the common intersection of two surfaces, e.g. 

of a cone and a cylinder. 

In such a case the curve 

M^ B is given by the equations 

of the two surfaces ;f say 





F(x,y,z) = 0,U 
<t>(x,y, z) =0.J 



(1) 



The tangent line to this 
curve, at any point on it, 

is the intersection of the 

two tangent planes, one 
for each surface, passing 
through the point. Ac- 
cordingly [by Art. 161, Equation (7)], the equations 
of the tangent line drawn through a point (x^ y 1} 2j) 
on the curve given by equations (1), are 



Fig. 95. 



04&.+(y-vi)^+(z-3i)^= 



(a? - a>{) 



die 1 

doc. 



dVi 



d»t 



By 1 l dz x 



(2) 



Equations (2), as may be seen on solving them for the values 



of the ratios 



x 



i y-yi 



z — z l 



z — z. 



, may be transformed into 



y—yi 



8Fd$_dFd$ 8Fd$_dFd$ dFd$ _dFd$ 
dy 1 dz 1 dz 1 dy 1 dz ± dx 1 dx 1 dz 1 dx 1 dy x dy 1 dx^ 



(3) 



In Fig. 95, APB is the curve, LP the tangent line, NP the normal plane. 



* This Article is supplementary to Arts. 159, 160. 

t Since the coordinates of any point on it satisfy the equation of each 
surface. 

% For example, see in Fig. 125 the curve B VB, which is the intersection 
of the sphere x 2 + y 2 + z 2 = a 2 and the cylinder x 2 + y 2 = ax. 



163.] 



SURFACES AXD TJVISTED CURVES. 



267 



From equations (3) and the principle quoted in the second 
footnote on page 265 the equation of the normal plane to the curve 
(1) at the point (x^ y 1} z ± ) is 



1; KdVi dz x dz x dyj v \dz ± dx x dx l dz l 



+ l l) \dx 1 dy l dv l dx 1 



0.(4) 



Xote. The expressions in the denominators in (3) may be 
expressed in the determinant forms: 



dF dF 

d<f> defy 


? 


dF dF 

dZ x ' BXy 

dcf> d<f> 
dzi dx 1 


? 


dF dF 
dab' dy 1 
d<f> d<£ 
dxi dy L 



EXAMPLES. 

1. Find the equations of the tangent line and the normal plane at the 
point (1,6, — 5) to the curve of intersection of the sphere x 2 + y 2 + z 2 — 
6 a; + 4 2 — 36 = and the plane x + 3 y — 22 = 29. 



Here F(;c, y, z) = x 2 + y 2 -\- z 2 -Q>x + ±z — 

0(x, y, 2) =z + 3?/ -22-29. 

Accordingly, 



3^_ 2x _ 6 , dF =2y , 



dx 



50 = 1 
dx 



dF. 
dz 
d± 
dz' 



22 + 4, 
-2. 



dy 
d± 
dy 

At the point (1, 6, — 5), x± = 1, y\ = 6, 2i =— 5 

The values of the above derivatives at (1, 6, — 5) are thus : 



4, 5^=12, ^=-6, 



50 _1 

9*i ' 



5*/i 
90 
5*/i 



= 3, 



50 
9«i 



-2. 



The equations of the tangent line at (1, 
(2), are thus : 



(X - 1)(- 4) + (y - 6) 12 + (2 + 5)(- 6) = 
(z-l)xl + (y-6)3 + («+5)(-2) = 
These simplifv to 

4 x - 12 ?/ + 6 2 + 98 = 0, | 

x + Sy — 2 s - 29 =0. J 



5), on substitution in result 



268 DIFFERENTIAL CALCULUS. [Ch. XVII. 

The equation of the normal plane to the curve at (1, 6, — 5), on substitu- 
tion in result (4) and simplification, is thus : 

3x + 7y + 12s + 15 = 0. 

2. Find the equations of the tangent plane and the equations of the nor- 
mal at the point (6, 4, 12) to the surface 

9 2 _ 4 X 2 _ 288 y. 

Also find the equations of the tangent line and the equation of the normal 
plane to the curve of intersection made with that surface at that point by 

(a) the plane Sx -2 y + z = 22 ; 
(&) the plane 4x + ?/-3s + 8 = 0. 

3. As in Ex. 2 at the point (5, 4, 2) on the surface 

y -2 + Z 2 = ix ^ 

taking for the planes of intersection : 

(a) 1 x-2y - z = 25, 
(6) 2x + Sy + z=24. 

4. As in Ex. 2 at the point (4, — 6, 3) on the surface 

4 s 2 + 9 ?/ 2 - 16 z 2 = 244, 
taking for the planes of intersection : 

(a) 3sc-2y-3z = 15, 
(6) x + 2y + 42 =4. 

5. Find the equations of the tangent line and the equation of the 
normal plane at the point (6, 4, 12) to the curve of intersection of the 
surfaces 

9z 2 -4z 2 = 288?/*| 

x 2 + y 2 + z 2 = 196. j 

6. Find the equations of the tangent line and the equation of the normal 
plane at the point (5, 4, 2) to the curve of intersection of the surfaces. 

y2 +Z 2 = 4 £ } f 

2 x 2 + 4 ij 2 + 3 z 2 = 126. 
1 N.B. For other examples, see Granville, Calculus, pp. 276, 278, 279. 



L26. J 



* See Ex. 2. t See Ex. 3. 



INTEGRAL CALCULUS. 

CHAPTER XVIII. 

INTEGRATION. 

N.B. If thought desirable, Art. 167 may be studied before Arts. 165, 166. 
(Remarks relating to the order of study are in the preface.) 

164. Integration and integral defined. Notation. In Chapter III. 
a fundamental process of the calculus, namely, differentiation, 
was explained. In this chapter two other fundamental processes 
of the calculus, each called integration, are discussed. The 
process of differentiation is used for finding derivatives and 
differentials of functions ; that is, for obtaining from a function, 
say F(x), its derivative F'(x), and its differential F'(x)dx. On 
the other hand the process of integration is used : 

(a) For finding the limit of the sum of an infinite number of 
infinitesimals which are in the differential form f(x) dx (see Art. 166) ; 

(b) For finding functions lohose derivatives or differentials are 
given ; that is, for finding anti-derivatives and anti-differentials 
(see Arts. 27 a, 167). 

Briefly, integration may be either (a) a process of summation, 
or (b) a process ivhich is the inverse of differentiation, and which, 
accordingly, may be called anti-differentiation. Integration, as a 
process of summation, was invented before differentiation. It 
arose out of the endeavor to calculate plane areas bounded by 
curves. An area was (supposed to be) divided into infinitesimal 
strips, and the limit of the sum of these was found. The result 
was the lohole (area) ; accordingly it received the name integral, 
and the process of finding it was called integration. In many 
practical applications integration is used for purposes of sum- 
mation. In many other practical applications it is not a sum 
but an anti-differential that is required. It will be seen in Art. 16(> 
that a knowledge of anti-differentiation is exceedingly useful in 
the process of summation. Exercises on anti-differentiation have 
appeared in preceding articles. 

269 



270 INTEGRAL CALCULUS. [Ch. XVIII. 

Note. The part of the calculus which deals with differentiation and its im- 
mediate applications is usually called The Differential Calculus, and the part 
of the calculus which deals with integration is called The Integral Calculus. 
With Leibnitz (1646-1716), the differential calculus originated in the problem 
of constructing the tangent at any point of a curve whose equation is given, 
This problem and its inverse, namely, the problem of determining a curve 
when the slope of its tangent at any point is known, and also the problem of 
determining the areas of curves, are discussed by Leibnitz in manuscripts 
written in 1673 and subsequent years. He first published the principles of 
the calculus, using the notation still employed, in the periodical, Acta 
Eruditorum, at Leipzig in 1684, in a paper entitled Nova methodus pro 
maximis et minimis, itemque tangentibus, quae nee fractas nee irrationales 
quantitates moratur, et singular e pro illis calculi genus. Isaac Newton 
(1642-1727) was led to the invention of the same calculus by the study of 
problems in mechanics and in the areas of curves. He gives some description 
of his method in his correspondence from 1669 to 1672. His treatise, 
Methodus fluxionum et serierum infinitarum, cum ejusdem applicatione ad 
curvarum geometriam, was written in 1671, but was not published until 1736. 
The principles of his calculus were first published in 1687 in his Principia 
(Philosophiae Naturalis Principia Mathematical . It is now generally 
agreed that Newton and Leibnitz invented the calculus independently of each 
other. For an account of the invention of the calculus by Newton and 
Leibnitz, see Cajori, History of Mathematics, pp. 199-236, and Cantor, 
Geschichte der Mathematik, Vol. 3, pp. 150-172. 

" There a?^e certain focal points in history toward which the lines of past 
progress converge, and from which radiate the advances of the future. Such 
was the age of Newton and Leibnitz in the history of mathematics. During 
fifty years preceding this era several of the brightest and acutest mathe- 
maticians bent the force of their genius in a direction which finally led to the 
discovery of the infinitesimal calculus by Newton and Leibnitz. Cavalieri, 
Eoberval, Fermat, Descartes, Wallis, and others, had each contributed to 
the new geometry. So great was the advance made, and so near was their 
approach toward the invention of the infinitesimal analysis, that both 
Lagrange and Laplace pronounced their countryman, Fermat, to be the true 
inventor of it. The differential calculus, therefore, was not so much an 
individual discovery as the grand result of a succession of discoveries by 
different minds." (Cajori, History of Mathematics, p. 200.) 

Also see the "Historical Introduction" in the article, Infinitesimal Cal- 
culus (Ency. Brit., 9th edition), and, at the end of that article, the list of 
works bearing on the infinitesimal method before the invention of the 
calculus. 

Notation. In differentiation d and D are used as symbols ; thus, 
df(x) is read " the differential of f(x) 9 " and Df(x) is read " the 



164, 165.] 



INTEGRATION. 



271 



derivative of /(«)." In integration, whether the object be sum- 
mation or anti-differentiation, the sign J is most generally used 
as the symbol ; thus, J f(x)dx is read "the integral off(x)dx."* 

Other symbols, viz. d~ 1 f{x)dx and D~ 1 f(x), are used occasionally 
(see Art. 167, Note 2). The quantity f(x) which appears " under 
the integration sign," as the mathematical phrase goes, is called 
the integrand. 

165. Examples of the summation of infinitesimals, These examples 
are given in order to help the student to understand clearly what 
the phrase " to find the limit of the sum of a set of infinitesimals 
of the form f(x)dx (i.e. a set of infinitesimal differentials)" means. 

(a) Find the area between the line y = mx, the x-axis, and the ordinates 

drawn to the line at 
x = a and x = b. 

Let PQ be the line 
whose equation is 
y = mx, OA = a, and 
OB =b. Draw the 
ordinates ^IPand BQ ; 
it is required to find 
the area APQB. 

Suppose that AB 
is divided into n equal 
parts each equal to Ax, 
so that 

n ■ Ax = b — a. 



Y 


p, 


p i 


p 


Pax 


G 


G 










\ & 







,- 


i i 


*i< 


\I. Z M v 


■1* 


3 X 



Fig. 96. 



Draw the ordinates at each point of division, Mi, M 2 , •••, M n -\ ; complete 
the inner rectangles PJii, Pi, M 2 , •••, P n -\B ; and complete the outer rectan- 
gles P\A, P<l2>I\, •••, QM n -\. The area APQB is evidently greater than the 
sum of the inner rectangles and less than the sum of the outer rectangles ; i.e. 

sum of inner rectangles < APQB < sum of outer rectangles. 



* The word integral appeared first in a solution of James Bernoulli (1654- 
1705), which was first published in the Acta Eruditorum in 1690. Leibnitz 
had called the integral calculus calculus summatorius, but in 1696 the term 
calculus integralis was agreed upon by Leibnitz and John Bernoulli (1667- 
1748). The sign ( was first used in 1675, and is due to Leibnitz. It is 
merely the long S which is the initial letter of summa, and was used by 
earlier writers to denote " the sum of." 



272 INTEGRAL CALCULUS. [Ch. XVIII. 

The difference between the sum of the inner and the sum of the outer rectangles 
is the sum of the rectangles PP X , P±P 2 , — , P" -1 Q. The latter sum is evidently 
equal to the rectangle QS, i.e. to CQ • Ax. This approaches zero when Ax 
approaches zero. Therefore APQB is the limit of the sum of either set of 
rectangles when Ax approaches zero. The limit of the sum of the inner 
rectangles will now be found. 



MA, 


x = a, 


and hence, 


AP = ma ; 


at Mi, 


x = a + Ax, 


and hence, 


M1P1 = m(a + Ax) ; 


at M 2 , 


x = a + 2 Ax, 


and hence, 


M 2 P 2 = m(a + 2 Ax) ; 











at M n -i, x = a + n — 1 Ax, and hence, M n -\P n -\ = m(a + n — 1 • Ax), 

.'. sum of inner rectangles 

= ma • Ax -f m(a + Ax)- Ax + m(a + 2 Ax) • Ax + ••• 



+ m{a + n — 1 • Ax) • Ax. 

.'. area APQB = lim Az=0 [ma Ax + m(a + Ax)Ax-i |_ m ( a -j- n _i . Ax) Ax] 

= lhn Az=0 m[a+(a + Ax)+(a+2 Ax)-\ \-{a + n — 1 • Ax)]Ax. 

Hence, on summation of the arithmetic series in brackets, 
area APQB = lim Aa ^o ^^{2 a + n^l ■ Ax}. 

On giving n Ax its value b — a, this becomes 
area APQB = lim Aj -o m ( b ~ a \ (b + a - Ax) 



-(1-1) 



Note 1. In this example the element of area, as it is called, is a rectangle 
of height y and width Ax when Ax is made infinitesimal, i.e. the element 
of area is y dx or mx dx in which dx = 0. (See Art. 27, Notes 3, 4, and 
Art. 67 a.) 

Note 2. It may be observed in passing that on taking the anti-differential 

of mx dx, namely ^- , substituting b and a in turn for x therein, and taking 

the difference between the results, the required area is obtained. 

Ex. Eind the limit of the sum of the outer rectangles when Ax approaches 
zero. 

(5) Find the area between the parabola y = x 2 , the x-axis, and the ordinates 
atx = a and x — b. 



165.] 



INTEGRATION. 



273 



Let LOQ be the parabola, OA = a, OB = b ; draw the ordinates AP 

and BQ ; the area APQB is 
required. As in the preceding 
problem, divide AB into n 
parts each equal to Ax, so that 

n Ax = b — a \ 

draw ordinates at the points 
of division, and construct the 
set of inner rectangles and 
the set of outer rectangles. 
As in (a), it can be seen that 
sum of inner rectangles < 
area APQB < sum of outer rectangles ; and also that 

(sum of outer rectangles) — (sum of inner rectangles) = CQ • Ax, 

which approaches zero when Ax approaches zero. Hence the area APQB is 
the limit of the sum of either set of rectangles when Ax approaches zero. 
The limit of the sum of the inner rectangles will now be found. 




At A, x — a, and hence, 

at Mi, x = a + Ax, and hence, 

at Mi, x = a + 2 Ax, and hence, 



AP=a 2 ; 
MiPi = (a + Ax) 2 ; 
M 2 P 2 = (a + 2 Ax)' 2 ; 



at M n _i, x = a + n — 1 • Ax, and hence, M n _iP n _i = (a + n — 1 • Ax) 2 . 
.'. sum of inner rectangles = a 2 Ax + (« + Ax) 2 Ax + (a + 2 Ax) 2 Acc + ••• 



+ (a + n - 1 • Az) 2 A£. 
area APQB = limAx^o{a 2 + (a + Ax) 2 + (a + 2 Ax) 2 + 



+ (a + n - 1 -Ax) 2 } Ax 



Now 
and 



= limA xi0 {wa 2 + 2 a Ax(l + 2 + 3 + — + w - 1) 
+ (Ax) 2 (l 2 + 2 2 + 3 2 + ... + n - l 2 )}Ax. 
1 + 2 + 3 + ... + w- 1 = J n(w - 1) ; 
12 + 2 2 + 3 2 + ... + ^^l 2 = | (m - 1)»(2 71 - 1).* 
,% area APQB = limAi^o w Ax {a 2 + aw Ax — a Ax + £ (n Ax) 2 
- | w (Ax) 2 + i (Ax) 2 }. 



* It is shown in algebra that the sum of the squares of the first n natural 
numbers, viz. I 2 , 2 2 , 3 2 , .-, ri 2 , is | n(n + 1)(2 n + 1). 



274 INTEGRAL CALCULUS. [Ch. XVIII. 

But n Ax = b — a, no matter what n and Ax may be. 

.-. area APQB = lim Ax =o (b - a){a 2 + a(6 - a) - a Ax + i (6 - a) 2 

+ i(&-a)Ax + i(Az) 2 } 

3 3' 

Note 1. In this example the element of area is a rectangle of height y 
and width Ax, when Ax becomes infinitesimal, i.e. the element of area is 
y dx, i. e. x 2 dx, in which dx = 0. 

Note 2. It may be observed in passing that the result (1) can be ob- 
tained by taking the anti-differential of x 2 dx, namely — , substituting b and 

o 

a in turn for x therein, and calculating the difference - — — • 

3 3 

Ex. Find the limit of the sum of outer rectangles. 

(c) Find the distance through which a body falls from rest in t\ seconds, 
it being known that the speed acquired in falling for t seconds is gt feet per 
second. [Here g represents a number whose approximate value is 32.2.] 

Note 1. If the speed of a body is v feet per second and the speed remains 
uniform, the distance passed over in t seconds is vt feet. 

Let the time ti seconds be divided into n intervals each equal to A£, so that 

nAt= t\. 

The speed of the falling body at the beginning of each of these successive 
intervals of time is 

0, g • At, 2 g • At, •■-, (n — l)g ■ At, respectively ; 

the speed of the falling body at the end of each successive interval of time is 

g ■ At, 2 g - At, 3 g • At, •••, ng • A£, respectively. 

For any interval of time the speed of the falling body at the beginning is 
less, and the speed at the end is greater, than the speed at any other moment 
of the interval. Now let the distance be computed which would be passed 
over by the body if it successively had the speeds at the beginnings of the 
intervals ; and then let the distance be computed which would be passed over 
by the body if it successively had the speeds at the ends of the intervals. 

The first distance = + g(At) 2 + 2 g(At) 2 + ••• + (w - 1)#(A0 2 
= [0 + 1 +2+... +O-l)MA0 2 
= ±n(n-l)g(At) 2 . 



165, 166.] INTEGRATION. 275 

The second distance =[1 -f- 2 + 3 -\ h n2g(Aty 

= %n(n + l)g(Aty. 

The actual distance fallen through, which may be denoted by s, evidently 
lies between these two distances ; i.e. 

J n(n - 1)<7(A0 2 < • < i »(n + 1)^(A0 2 . 

On putting t\ for its equal, n At, this becomes 

igh 2 - igh • At<s<igh* + igh • At. 

On letting At approach zero these three distances approach equality, and 

hence s = J gt{ 1 . 

Note 2. For two other examples see Art. 166, Note 4. 



Pn-& 



166. Integration as summation. The definite integral. It will 
now be shown, geometrically, how integration is a process of sum- 
mation. Let f{x) denote any function of x which is continuous 

from x = a to x = b and geometri- 
cally representable. Let its graph 
be the curve K whose equation is 
accordingly .. . 

Suppose that OA = a and OB = b, 
and draw the ordinates AP and BQ. 
Divide AB into n parts, each equal 
to Ax ; accordingly, 




7i Ax = b — a. 



(i) 



At the points of division erect ordinates, and construct inner 
and outer rectangles as in Art. 165 (a), (6). It can be shown, as 
in the examples in Art. 165, that the difference between the set of 
the inner rectangles and the set of the outer rectangles is CQ • Ax 
(CQ being equal to BQ — AP), a difference which approaches 
zero when Ax approaches zero. The area APQB lies between 
these sets and evidently is the limit of the sum of either set of 
rectangles when Aa; approaches zero. The limit of the sum of 
inner rectangles will now be found. 



276 INTEGRAL CALCULUS. [Ch. XVIII. 

At A, x = a, and hence, AP =f(a) ; 

at M lt x = a + Ax, and hence, M X P Y = f(a + Ax) ; 

at M 2 , x = a + 2 Ax, and hence, M 2 P 2 =/(« + 2 A#) ; 

at Jf n _ 1? x = b — Ax, and hence, M n __ 1 P n _ 1 =f(b — Ax). 

.-. area APQB = lim Ax=y) 
\f(a)Ax+f(a + Ax)Acc +/(a + 2 Aa;)Aaj + ••• +/(& - Aa?)AxJ. (2) 

The second member, which is the sum of the values, infinite in 
number, that f(x)Ax takes when x increases from a to & by equal 
infinitesimal increments Ax, may be written (i.e. denoted by) 

x=b 

lim^ ^f(x)Ax* 

x=a 

It is the custom, however, to denote the second member of (2) 
by putting the old-fashioned long S before f(x)dx and writing at 
the bottom and top of the # respectively the values of x at which 
the summation begins and ends ; thus 

f{oc)doc; or, more briefly, I f(oc)dx, (3) 

x=a «/a 

This symbol is read "the integral of f(x)dx between the limits 
a and b," or " the integral of f(x)dx from x = a to x = &." 

Note 1. The numbers a and b are usually called the lower and upper 
limits of x. It would be better, perhaps, not to use. the word limit in this 
connection, but to say "the initial and final values of x," or simply, "the 
end-values of x. " f 

Note 2. The infinitesimal differential f(x)dx is called an element of 
the integral. It is the area of an infinitesimal rectangle of altitude f{x) and 
infinitesimal base dx. 



* The latter part of this symbol denotes, and is to be read, "the sum of 
all quantities of the type" [or "form"] ii f(x)Ax, from x = a to x = 6" 
[or " between x = a and x = b "]. 

t Joseph Fourier (1768-1830) first devised the way shown in (3) of indi- 
cating the end-values of x. 






166.] INTEGRATION. 277 

Note 3. It is not necessary that the infinitesimal bases, i.e. the increments 
Ax of x, he all equal ; hut for purposes of elementary explanation it is some- 
what simpler to take them as all equal. (See Lamb, Calculus, Arts. 86, 87, 
and the references in Art. 167, Note 5 ; also Snyder and Hutchinson, Calculus, 
Art. 150.) 

Note 4. For the calculation of ( e x clx and \ sinxdx by the process 
shown in Art. 165, see Echols, Calculus, Art. 125. 

The sum in brackets in (2) will now be calculated, and then its 

limit, which is indicated by the symbol (3), will hs found. 

Let the anti-differential (Art. 27 a) of f(x)dx* be denoted by 

<f> (x) : that is, let _, N , _ , , N 

vw ' ' f{x)dx = d<l>(x). 

Then, by the elementary principle of differentiation (see Art. 22, 
Note 3) for all values of x from a to b, 

<H* + **)-4>(x) ssf(!e) + e) (4) 

in which e denotes a function whose value varies with the value 
of x, and which approaches zero when Ax approaches zero. On 
clearing of fractions and transposing, (4) becomes 

J(x) Ax = <f> (x + Ax) — <f> (x) — e • Ax. (5) 

On substituting a, a + Ax, a + 2 Ax, •••, b — Ax in turn for x in 
(5), and denoting the corresponding values of e by e 1} e 2 , e 3 , •••, e n , 

respectively, there is obtained: 

f(a) Ax = <f>(a+ Ax) — cf> (a) — e 1 • Ax, 

f(a + Ax) Ax = <j>(a + 2 Ax) — <f> (a + Ax) — e 2 • Ax, 

f(a + 2 A x) Ax = cf> (a + 3 Ax) -<f>(a + 2 Ax) - e 3 • Ax, 



f(b-Ax)Ax = <f>(b) -cf>(b-Ax) - e n - Ax. 

* If /(x) is a continuous function of x, /(x) dx has an anti-differential. For 
proof see Picard. Traite d" 1 Analyse, t. I. No. 4 ; also see Echols, Calculus, 
Appendix, Note 9. 



278 INTEGBAL CALCULUS. [Ch. XVIII. 

Addition gives 

/(a) Ax +/(a + Ax) Ax +/(a + 2 Ax) Ax H \-f(b — Ax) 

= 4>(b)- <f>(a) - (e x + e 2 + e 3 + ••• + c.) Ax. (6) 

On taking the limit of each member of (6) when Ax approaches 



zero, 

J /(a?) dx = <j>(b)-<f> (a) - lim Axi0 0i + e 2 + • • • + e„) Ax. (7) 

Let e 1 be one of the e's which has an absolute value E not less 
than any of the others ; then evidently 

0i + e 2 H h e n ) Ax < nEAx; 

i.e. by (1), (ei + e 2 + ••• + <QAx< (b-a)E. 

Hence, lim^^ (e x -+- e 2 H + e») Ax = 0, since i2 approaches zero 

when Ax approaches zero ; and therefore, 

JV(sc) to = <K&) - <K«). (8) 

That is, expressing (8) in words : The integral I f(x) dx, which 

is the limit of the sum of all the values, infinite in number, that 
f(x) dx takes as x varies by infinitesimal increments from a to b, is 
obtained by finding the anti-differential, <j>(x), of f(x)dx, and then 
calculating <£ (6) — <f> (a). 

Note 5. Many practical problems, such as finding areas, lengths of curves, 
volumes and surfaces of solids, and so on, can be reduced to finding the limit 
of the sum of an infinite number of infinitesimals of the form f(x) dx. (See 
Arts. 181, 182, 207-212.) As has been seen above, the anti-differential 
of f{x) dx is of great service in determining this limit ; accordingly, con- 
siderable attention must be given to mastering methods for finding anti- 
differentials. 

Note 6. The process of finding the anti-differential of /(x) dx is nearly 
always more difficult than the direct process of differentiation, and frequently 
the deduction of an anti-differential is impossible. When the anti-differential 
of f(x) dx cannot be found in a finite form in terms of ordinary functions, 
approximate values of the definite integral can be found by methods dis- 
cussed in Chapter XXII. The impossibility of evaluating the first member of 
(8) in terms of the ordinary functions has sometimes furnished an occasion 
for defining a new function, whose properties are investigated in higher 
mathematics. (On this point see Snyder and Hutchinson, Calculus, Art. 123, 



166.] 



INTEGRATION. 



279 



foot-note.) For instance, the subject of Elliptic Functions arose out of the 
study of what are called the elliptic integrals (see Art. 209, Ex. 4, Art. 199, 
Note 4, Art. 192, Note 4). 

(The ordinary elementary functions can be defined by means of the 
calculus, and their properties thence developed.) 

Note 7. At the beginning of this article the principle was enunciated 
that the area bounded by a smooth curve PQ (Fig. 98), the x-axis, and a pair 
of ordinates, is the limit of the sum of certain inner, or outer, rectangles 
constructed between the ordinates. The student can easily show that this 
principle holds for the smooth curves in Figs. 99 a, 6, c. 




B X 

Fig. 99 a. 



Note 8. This article shows that a definite integral may be represented 
geometrically as an area. For a general analytical exposition of integration 
as a summation, see Snyder and Hutchinson, Calculus, Art. 148. Their 
exposition depends on Taylor's theorem (Art. 150). Also see the references 
mentioned in Art. 167, Note 5. 

Ex. Show that the calculus method of computing the area in Fig. 99 c 
bounded by PMNBQ, AB, AP, and BQ really gives area^lPJf + area, RQB 
— area MNB. 

[As a point moves along the curve from Pto Q, dx is always positive. In 
ABM y is positive, in MNB negative, in BQB positive. Accordingly, the 
elements of area,/(x) dx or y dx, are positive in A BM and BQB, and negative 
in MNB.~] 

EXAMPLES. 



N.B. The knowledge already obtained in Chapter IV. about anti-differen- 
tials is sufficient for the solution of the following examples. It is advisable 
to make dravnngs of the curves and the figures whose areas are required. 



1. Find the area between the cubical parabola 
x-axis, and the ordinates for which x = 1, x = 3. 



x 3 (Fig., p. 462), the 



280 INTEGRAL CALCULUS. [Ch. XVIII. 

According to (3) and (8), the area required = f x s dx 

= ^ + c-(i + c) 

= 20 sq. units of area. 

2. Find the area between the curve in Ex. 1, the x-axis, and the ordi- 
nates for which x = — 2, as = 3. Ans. 16} sq. units. 

3. Explain the apparent contradiction between the results in Exs. 1, 2. 

4. Find the actual number of square units in the figure whose boundaries 
are given in Ex. 2. Ans. 24i sq. units. 

5. Find the area between the parabola 2 y = 7 x 2 , the x-axis, and the 
ordinates for which : (1) x = 2, x = 4 ; (2) x = — 3, x = 5. 

Ans. (1) 65i sq. units ; (2) 177£ sq. units. 
N.B. A table of square roots will save time and trouble. 

6. Find the area between the parabola y 2 = 8 x, the x-axis, and the 
ordinates for which : (1) x = 0, x = 3 ; (2) x = 2, x = 7. 

Ans. (1) 9.798 sq. units ; (2) 29.59 sq. units. 

7. Find the area of the figure bounded by the parabola y 2 = 6 x and 
the chord perpendicular to the x-axis at x = 4. Ans. 26.128 sq. units. 

8. Find, by the calculus, the area bounded by the line y = 3 x, the 
x-axis, and the ordinate for which x = 4. Ans. 24 sq. units. 

9. (1) Find, by the calculus, the area of the figure bounded by the line 
y = 3 x, the x-axis, and the ordinates for which x = 4, x = — 4. (2) How 
many sq. units of gold leaf are required to cover this figure ? 

Ans. (1) 0; (2) 48 sq. units. 

10. (1) Find the area between a semi-undulation of the curve y — sin x 
and the x-axis. (2) Find the area of the figure bounded by a complete 
undulation of this curve and the x-axis. (3) How many sq. units of gold- 
leaf are required to cover this figure. Ans. (1) 2 ; (2) ; (3) 4. 

11. Compute the area enclosed by the parabola y 2 = 4x and the lines 
x = 2, x = 5. Ans. 22.27 sq. units. 

12. Compute the area enclosed by the parabola y = x 2 and the lines 
y = 1, y = 4. Ans. 9±- sq. units. 

13. Find the area between the parabolas x 2 ~y and y 2 = 8 x. 

Ans. 2| sq. units. 

14. Find the area between the curves : (1) y 2 = x and y 2 = x 3 ; (2) x 2 = y 
and y 2 = x 3 . (Make figures. ) Ans. (1) T 4 5 sq. units; (2) ^ sq. units. 

15. Find the area bounded by the curves in Ex. 14 (2) and the lines 
x = 2, x = 4. Ans. 8.129 sq. units. 

N.B. Art. 181 may be taken up now. 



166,167.] INTEGRATION. 281 

167. Integration as the inverse of differentiation. The indefinite 
integral. Constant of integration. Particular integrals. In many 
cases there is required, not the limit of the sum of an infinite 
number of infinitesimals of the form f(x)dx, but the function 
whose derivative or differential is given. The following is an 
instance from geometry. When a curve's equation, y=f(x), is 
known, differentiation gives the slope at any point on the curve 

in terms of the abscissa x, namely, -^-=f'{x) (Art. 24). On the 

ClXi 

other hand, if this slope is given, integration affords a means of 
finding the equation of the curve (or curves) satisfying the given 
condition as to slope. Again, an instance from mechanics : if a 
quantity changes with time in an assigned way, differentiation 
determines the rate of change for any instant (Art. 25). On the 
other hand, if this rate of change is known, integration provides 
a means for determining the quantity in terms of the time. (See 
Art. 22, Notes 1, 2, and Art. 27 a.) 



EXAMPLES. 

Ex. 1. The slope at any point (x, y) of the cubical parabola y = x 3 is 3x 2 ; 

that is, at all points on this curve, -^ = 3 x 2 and dy = 3 x 2 dx. 

dx 

Now suppose it is known that a curve satisfies the following condition, 

namely, that its slope at any point (x, y) is 3 x 2 ; i.e. that for this curve, 

^ = 3 x 2 , (whence, dy = 3 x 2 dx). 
dx 

Then, evidently, y = x 3 + c, 

in which c is a constant which can take any arbitrarily assigned value. This 
number c is called a constant of integration ; its geometrical meaning is 
explained in Art. 99. Since c denotes any constant, there is evidently an 
infinite number of curves (cubical parabolas, y = x 3 + 2, y = x s — 10, y — x 3 
+ 7, etc., etc.) which satisfy the given condition. If a second condition is 
imposed, the constant c will have a definite and particular value. For 
instance, let the curve be required to pass through the point (2, 1). Then, 
1 = 2 3 -+- c ; whence c = — 7, and the equation of the curve satisfying both 
the conditions above is y = x 3 — 7. (Also see Ex. 17, Art. 37.) 

2. Suppose that a body is moving in a straight line in such a way that 
(the number of units in) its distance from a fixed point on the line is always 



282 INTEGRAL CALCULUS. [Cii. XVIII. 

(the number of units in) the logarithm of the number of seconds, t say, since 

the motion began ; i. e. so that 

s — log t. 

Then, the speed, — = 1. and ds=—- 

* dt t t 

Now suppose it is known that at any time after the beginning of its 

motion, after t seconds say, the speed of a moving body is -; i.e. that 

t 

/ whence, ds =- 



dt t \ t 

Then, evidently, s = log t + c, 

in which c is an arbitrary constant. If a second condition is imposed, the 
constant c will take a definite value. For instance, let the body be 4 units 
from the starting-point at the end of 2 seconds, i. e. let s = 4 when t = 2. 

lhen 4 = log 2 + c ; whence c = 4 - log 2, 

and s = log t + 4 — log 2. 

3. In Ex. 1 determine c so that the cubical parabola shall go through 
(a) the point (0, 0); (6) the point (7, -4); (c) the point (-8, 2); (d) the 
point (h, 7c). Draw the curves for (a), (b), (c). 

4. Find the curves for which the slope at any point is 4. Determine 
the particular curves which pass through the points (0, 0), (2, 3), (—7, 1), 
respectively. Draw these curves. 

5. Find the curves for which (the number of units in) the slope at 
any point is 8 times (the number of units in) the abscissa of the point. 
Determine the particular curves which pass through the points (0, 0), (1, 2), 
(2, 3), (—4, 2), respectively. Draw these curves. 

6. How are the curves in Exs. 4, 1, 3, 5, respectively, affected when 
the constants of integration are changed ? 

7. If at any moment the velocity in feet per second at which a body 
is falling is 32 times the number of seconds elapsed since it began to fall from 
rest, what is the general formula for its distance, at any instant, from a point 
on the line of fall ? 

In this instance, — = 32 t, (whence, ds = 32 t dt). 

Hence s = 16 t 2 + c. 

8. In Ex. 7, at the end of t seconds what is the distance measured 
from the starting-point ? What is the distance at the end of 2 seconds ? of 
4 seconds ? of 5 seconds ? What are the distances, in these respective dis- 
tances, measured from a point 10 feet above the starting-point ? If at the 
time of the beginning of fall, the body is 20 feet below the point from which 



167.] INTEGRATION. 283 

distance is measured, what is its distance below this point at the end of t 
seconds ? Explain the meaning of the constant of integration in the general 
formula derived in Ex. 7 ? Derive the results in Ex. 8 from this general 
formula. 

Suppose that d<f>(x) = f(x)dx-, (1) 

then also (Art. 29), d \4>(x) + c j = f(x)dx, (2) 

in which c is any constant. Hence, if <f>(x) is an anti-differential 
of f(x)dx, <j>(x) + c is also an anti-differential of f(x)dx. That is, 
if d<j>(x) = f(x)dx, 

then $f(x)dx = <Kas) + c, (3) 

in which c is an arbitrary constant. Thus the anti-differential of 
f(x)dx is indefinite, so far as an added arbitrary constant is con- 
cerned. (This has already been pointed out in Art. 29, Note 6.) 
On this account the anti-differential is called the indefinite integral. 
The arbitrary constant is called the constant of integration. The 
indefinite integral is often called the general integral. If the 
constant of integration be given a particular value, as \, —2, 
100, etc., the integral is called a particular integral. For instance, 

the indefinite, or general, integral of x 5 dx, i.e. I x*dx is \x Q + c\ 

and particular integrals of x*dx are i x G -f 5, i x Q — 11, etc. 

9. Name the indefinite (or general) integrals and the particular integrals 
appearing in Exs. 1-8. 

10. How many particular integrals (anti-differentials) can a function 
have ? TVhat must the difference between any pair of them be ? 

Xote 1. It should be noted that the indefiniteness in the integral does 
not extend to the terms involving the variable. For instance, 



f 



(x + l)c7z = ix 2 + x+ c, 

and f (x -f \)dx = ( (x + l)d(x + 1)* = \(x + l) 2 + Tc = \x* + x + l + k\ 
thus the terms involving x are the same. 

Xote 2. The origin of the words integral and integration has been 
indicated in Art. 164. It is, in a measure, to be regretted that the term 

integral and the symbol \ , which both imply summation, should also be 
used to denote an anti-differential. In accordance with the fashion in vogue 

* Since d(x + 1) = dx. 



284 INTEGRAL CALCULUS. [Ch. XVIII. 

in trigonometry for denoting inverse functions {e.g. sin x and sin -1 x for sine 
of x and anti-sine, or inverse sine, of x, respectively *) the anti-derivative 
of/(x) and the anti-differential of f(x) dx are sometimes denoted by D~ 1 f{x) 

and d~ 1 f(x)dx respectively. Thus \f(x)dx, d^f^dx, and D" 1 /^), are 
equivalent. 

Note 3. If d<f> (x) =f(x) dx, then (Art. 166) f f(x) dx - <f> (x) - <f>(a). 

Ja 

If the upper end- value x is variable, and the lower end-value a is arbitrary, 
then this integral is indefinite and of the form <f> (x) + c. Accordingly, the 
indefinite integral may be regarded as in the form of a definite integral whose 
upper end-value is the variable, and whose lower end-value is arbitrary. 

Note 4. Result (8), Art. 166 for the area of APQB (Fig. 98) can also be 
derived by a method which is founded on the notion of the indefinite integral. 
For instance, see Todhunter, Integral Calculus, Art. 128, or Murray, Integral 
Calculus, Art. 13. 

Note 5. Keferences for collateral reading on the notions of integra- 
tion, definite integral, and indefinite integral. Gibson, Calculus, §§ 82, 110, 
124-126 ; Williamson, Integral Calculus, Arts. 1, 90, 91, 126 ; Harnack, 
Calculus (Cathcart's translation), §§ 100-106 ; Echols, Calculus, Chap. XVI. ; 
Lamb, Calculus, Arts. 71, 72, 86-93. 

168. Geometric or graphical representation of definite integrals. 
Properties of definite integrals. It has been seen (Art. 166) that 
if PQ (Fig. 98) is the curve whose equation is 



then the integral J f(x) 



dx 



gives the area bounded by the curve, the x-axis, and the ordinates 
for which x = a and x = b respectively. Accordingly, the figure 
thus bounded may be said, and may be used, to represent the 
integral graphically. Hence, in order to represent an integral, 

cf> (x) dx say (no matter whether this integral be an area, or a 

length, or a volume, or a mass, etc.), draw the curve whose 
equation is y = cf>(x), and draw the ordinates for which x = l and 
x = m respectively. The figure bounded by the curve, the a>axis, 
and these ordinates, is the graphical representative of the integral, 
and (Art. 166) the number of units in the area of this figure is the 
same as the number of units in the integral. 

* See Art. 12, Note. 



167, 168.] 



INTEGRATION. 



285 



The following properties of definite integrals are important. Prop- 
erties (b) and (c) are easily deduced by using the graphical 
representatives of the integrals. 

(a) If dcf>(x)=f(x)dx, then (Art. 166) 

f(x) dx = cf> (b) — cf> (a) and j f(x) dx = <£ (a) — <£ ( 6) ; 

and hence, I f{x)dx = — I f(x)dx. 

Therefore, if the end-values of the variable in an integral be 
interchanged, the algebraic sign of the integral will be changed. 

Ex. Give several concrete illustrations of this property. 

f(x)dx— I f(x)dx-\- j f(x)dx, whatever c may be. 

Draw the curve y=f(x), and draw ordinates AP, BQ, CB, for 
which x = a, x = b, x== c, respectively: Then : 





c x 



Fig. 100 
In Fig. 100 a, 



C f(x)dx = area APQB 
= area APEC + area CRQB 

=jj(x)dx+£f{x)dx. 



Fig. 100 b. 
In Fig. 100 b, 
C f(x)dx = area APQB 

%J a 

= area ^Pi^C - area £Qi20 
== Cf(x)dx- ff(x)dx 

= Cf(x)dx+ Cf(x)dx. 



286 INTEGRAL CALCULUS. [Ch. XVIII 

Similarly, it can be shown that 

Cf(x)dx= Cf(x)dx + f d f(x)dx + .~+ Cf(x)dx + Cf(x)dx. 

x/a J a Jc Jit Jl 



That is, a definite integral can be broken up into any number of 
similar definite integrals that differ only in their end-values. 
(Similar definite integrals are those in which the same integrand 
appears.) 

Ex. 1. Prove the principle just enunciated. 

Ex. 2. Give concrete illustrations of the principles in (b). 

(c) Hie mean value of f(x) for all values of x from a to b. 
(That is, the mean value of f(x) when x varies continuously 



/ 



O A C B X 

Fig. 101 a. 









Y 


• 












( 


} 


I 






B/R 


M 


X 












A 


L L 







C 


1 B X 



Fig. 101 b. 



from a to b.) Draw the curve y=f(x), and at A and B erect 
the ordinates for which x = a and x = b respectively. Then 



f f(x)dx = area APQB. 



Now, evidently, on the base AB there can be a rectangle whose 
area is the same as the area of APQB. Let ALMB, which has 
an altitude CR, be this rectangle ; then 

I f(x) dx = area ALMB = area AB • CR 

= (b-a)> length CR. (1) 



168, 169.] INTEGRATION. 287 

The length CR is said to be the mean value of the ordinates 
f(x) from x = a to x = b. Hence, from (1), 

Mean value of /(a?) from j = j a /(^) ^ # 

In words, the mean value off(x) when x varies continuously from 
a to b, is equal to the integral of f(x)dx from the end-value a to 
the end-value b, divided by the difference between these end-values. 

EXAMPLES. 

1. Make a graphical representation of each of the integrals appearing 
in Exs. 2-5 below. 

2. Find the mean length of the ordinates of the parabola y = x 2 from 

x = 1 to x = 3. rs 

\ x 2 dx 

Mean length = ^ = 4i 

5 3-1 3 

3. In the parabola y — x 2 } find the mean length of the ordinates of the 
arc between x = and x = 2 ; and find the mean length of the ordinates 
from x = — 2 to x = 2. Explain, with the help of a figure, why these mean 
lengths are the same. 

4. In the cubical parabola y = x 3 . 

5. In the line y = 4 x. 

169. Geometric (or graphical) representation of indefinite integrals. 
Geometric meaning of the constant of integration. If 

d<f>(x) =f(x) dx, 

then (Art. 167) Cf(x) dx = <f>(x) + c, (1) 

in which c is an arbitrary constant. Draw the curve 

y = 4>(x) ; (2) 

let AB be the curve. Give c the particular values 2 and 10, and 
draw the curves, y = <j>{x) + 2 (3) 

and y = cf>(x) + 10. (4) 

*For clear proof that this is the mean value, see Art. 213, where the 
topic of mean values is more fully discussed, and Echols, Calculus, Art. 150 
(and Arts. 151, 152). 



288 



INTEGRAL CALCULUS. 



[Ch. XVIII. 



Let CD and EF be these curves. In the case of each one of the 
curves obtained by giving 
particular values to c, 

and hence, at points having 
the same abscissa the tan- 
gents to these curves have 
the same slope, and, accord- 
ingly, are parallel. For in- 
stance, on each curve, at 
the point whose abscissa is 
m the slope of the tangent is /(m). 

Moreover, the distance between any two curves obtained by 
giving c particular values, measured along any ordinate, is always 
the same. For, draw the ordinates KR and ST at x = m and 
x = n, respectively, as in the figure. Then, by Equations (3) 

and (4), MK= 0(m) + 2 ; NS = 0(n) + 2 ; 

and 
MR = <£(m) -f- 10; NT = <f>(n) + 10. 




Fig. 102. 



Hence 



KR = S, 



and ST =8. 



Accordingly, the graphical representation of the indefinite integral, 

I f(x) dx, consists of the family of curves, infinite in number, 

whose equations are of the form y = <j>(x) + c, and which are 
severally obtained by giving c particular values ; and the effect of 
changing c is to move the curve in a direction parallel to the 
?/-axis. (Also see Art. 29, Note 2.) 

Ex. 1. How many different values can be assigned to c? How many- 
particular integrals are included in the general integral ? How many different 
curves can represent the indefinite integral ? 

Ex. 2. Write the equations of several curves representing each of the 

following integrals, viz. : jxda;, ( x 2 dx, \Sx dx, \Sdx, f (2 x + 5) dx. 
Draw the curves. 



169, 170.] 



INTEGRATION. 



289 



170. Integral curves. If dcf>(x) =f(x)dx, 
then (Art. 166) ("/(*) dx = <f> (*) - <t> (0). 

The curve whose equation is 

y = <f> (x) - <f> (0), i.e. y = f X f(v) doc, (1) 

which is one of the particular curves representing y = <f> (x) + c 
(see Art. 169), is called the first integral curve for the curve y =f(x). 
Since the area of the figure bounded by the curve y =f(x), the 
a>axis, and the ordinates at x = and x = x, is cf>(x) — <£(0) (Art. 
166), the number of units of length in the ordinate at the point of 
abscissa x on the curve (1), is the same as the number of units 
of area in this figure. Accordingly, if the first integral curve of 
a given curve be drawn, the area bounded by the given curve, the 
axes, and the ordinate at any point on the #-axis, can be obtained 
merely by measuring the length of the ordinate drawn from the 
same point to the integral curve. Consequently, it may be said 
that this ordinate graphically represents the area, and thus, the 
integral. 

f(x) is the derived or differential curve 

(2) 



Note 1. The original curve y 
of curve (1). 

Ex. For instance, for the line y = J x + 3 ; 



j> 



x + 3) dx = i z 2 + 3 x, 



the first integral curve of curve (2) is the parabola y = i x 2 + 3 x. (3) 

These two curves are shown 
here. If M be any point on the 
x-axis, and 03I=m units of length, 
and the ordinate MLG be drawn, 



(the number of units of length 
in itf"6r) = (the number of units of 
area in OKLM). 

Tor, length MG, by (3), is \ m* 
+ 3w» ; and 

area OKLM 
= f m (£* + 3 ) dx = J m 2 + 3 m. 




Fio. 103. 



290 INTEGRAL CALCULUS. [Ch. XVIII. 

Just as a given curve — it may be called the original or the 
fundamental curve — has a first integral curve, this first integral 
curve also has an integral curve. The latter curve is called the 
second integral curve of the fundamental curve. Again, the second 
integral curve has an integral curve ; this is said to be the third 
integral curve of the fundamental curve. On proceeding in this 
way a system of any number of successive integral curves may 
be constructed belonging to a given fundamental curve. 

Note 2. The integral curve can be drawn mechanically from its funda- 
mental by means of an instrument called the integraph, invented by a 
Russian engineer, Abdank-Abakanowicz. 

Note 3. Integral curves are of great assistance in obtaining graphical 
solutions of practical problems in mechanics and physics. For further in- 
formation about integral curves and their uses and the theory of the integraph, 
and for other references, see Gibson, Calculus, §§ 83, 84 ; Murray, Integral 
Calculus, Art. 15, Chap. XII., pp. 190-200 (integral curves), Appendix, 
Note G (on integral curves), pp. 240-245; M. Abdank-Abakanowicz, Les 
Integraphes : la courbe integrate et ses applications (Paris, Gauthier-Villars), 
or BitterlVs German translation of the same, with additional notes (Leipzig, 
Teubner). Also see catalogues of dealers in mathematical and drawing 
instruments. 

EXAMPLES. 

1. Show that, for the same abscissa, the number of units of length in 
the ordinate of the fundamental curve is the same as the number of units in 
the slope of its first integral curve. 

2. Does the first integral curve belong to the family of curves referred to 
in Art. 99 ? 

3. Show how the members of the family of curves in Art. 169 may be 
easily drawn when an integraph is available. 

4. Write the equations of the first, second, and third integral curves 
of the following curves : (a) y = x ; (b) y = 2 x + 5 ; (c) y = sin x ; (d) y = e, x . 
Draw all these fundamental and integral curves. Can the curve x 3 y = 1 be 
treated in a similar manner ? 

5. Find and draw the curve of slopes for each of the curves (a), (6), 
(c), (d), Ex. 4. Then find and draw the first, second, and third integral 
curves of each of these curves of slope. 

171. Summary. The two processes of the infinitesimal calculus, 
namely, differentiation and integration, have now been briefly 
described. 



170, 171.] INTEGRATION. 291 

The process of differentiation is used in solving this problem, 
among others : the function of a variable being given, find the 
limiting value of the ratio of the increment of the function to the 
increment of the variable when the increment of the variable 
approaches zero (Art. 22). This problem is equivalent to finding 
the ratio of the rate of increase of the function to the rate of 
increase of the variable (Art. 26). If the function be represented 
by a curve, the problem is equivalent to finding the slope of the 
curve at any point (Art. 24). 

The process of integration may be regarded as either : 

(a) a process of summation ; or 

(6) a process which is the inverse of differentiation. 

Integration is used in solving both of the following problems, 
viz. : 

(1) To find the limit of the sum of infinitesimals of the form 
f(x) dx, x being given definite values at which the summation 
begins and ends (Arts. 164-166) ; 

(2) To find the anti-differential of a given differential fix) dx 
(Art. 167). 

Problem (1) is equivalent to finding a certain area; problem 
(2) is equivalent to finding a curve when its slope at every point 
is known. 

In solving problem (1) the anti-differential of f(x) dx is required 
(Art. 166). Hence, in both problems (1) and (2) it is necessary to 
find the anti-differentials of various functions of the form fix) dx. 
Chapters XIX. and XXI. are devoted to showing how anti-differ- 
entials may be found in the case of several of the comparatively 
small number of functions for which this is possible. It may be 
stated here^that, in general, integration is more difficult than the 
direct process of differentiation. 



CHAPTER XIX. 

ELEMENTARY INTEGRALS. 

172. In this chapter the elementary or fundamental integrals 
(anti-differentials) are obtained, and some general theorems and 
particular methods which are useful in the process of anti-differ- 
entiation are described. There is one general fundamental process 
(Art. 22) by which the differential of a function can be obtained. 
On the other hand, there is no general process by which the anti- 
differential of a function can be found.* The simplest integrals, 
which are given in Art. 173, are discovered by means of results 
made known in differentiation. 

In Art. 174 certain general theorems in integration are deduced. 
Two particular processes, or methods, of integration which are 
very serviceable and frequently used, are described in Arts. 175, 

176. A further set of fundamental integrals is derived in Art. 

177. When f(x) is a rational fraction in x, the anti-differential 
of f(x)dx may be found by means of the results in Arts. 173, 177; 
for this reason examples involving rational fractions are given in 
Art. 178. The integration of a total differential is considered in 
Art. 179. 

So far as finding anti-differentials is concerned, this is the most 
important chapter in the book. The student is strongly recom- 
mended to make himself thoroughly familiar with the chapter 
and to work a large number of examples, so that he can apply its 
results readily and accurately. The list of formulas, I. to XXVI. 
(Arts. 173, 177), should be memorized. Every function, f(x)dx, 
whose integral can be expressed in finite form in terms of the 
functions in elementary mathematics, is reducible to one or more 
of the forms in this list. It is often necessary to make reductions 
of this kind. A ready knowledge of these forms is not only useful 

* There is a general process by which the value of a definite integral can 
be found approximately, as described in Art. 193. 

292 



172, 173.] ELEMENTARY INTEGRALS. 293 

for integrating them immediately when presented, but is also a 
great aid in indicating the form at which to aim, when it is neces- 
sary to reduce a complicated expression. 

173. Elementary integrals. The following formulas in integra- 
tion come directly from the results in Arts. 37-55, and can be 
verified by differentiation. Here u denotes a function of any 
variable, and c, c , c 1} denote arbitrary constants. 

I. \ u n du = — — - + c, m which n is a constant. 
J n + 1 

Note 1. This result is applicable in the case of all constant values of w, 
excepting n = — 1. The latter case is given in II. 

II. f — = log u + co = log u + log c = log C2*. 
J u 

Note 2. The various ways in which the constant of integration can 
appear in this integral, should be noted. 

Note 3. Formula II. can also be derived by means of I. (See Murray, 
Integral Calculus, p. 37, foot-note ) 

III. (a u du=-^- + c. 
J log a 

IV. (e u du = e tl + c. 

V. ( sin u du = - cos u + c. 

VI. ( cos u du - sin u + c. 

Til. J sec 2 w du = tan w + c. 

VI II. j esc 2 «£ <?t* = — cot w + c, 

IX. j sec u tan t*e£w = sect* 4- c. 

X. j cscw cot udu= — esc w + c. 

XI. f ^ = sin- 1 m + c = - cos 1 u + c t . 

J Vi _ M a 

[Remark. By trigonometry sin" 1 w = - cos -1 w + 2 ktt + - . See Art. 107, 
Ex. 10 and Note 1.] 2 



294 INTEGRAL CALCULUS. [Ch. XIX. 

XII. f _^*_ = tan- 1 u + c. 

J 1 + u 2 

XIII. f du = sec 1 u + c. 

J uV u 2 - 1 

XIY. f ^ =YCTH- 1 U + C. 

Note 4. Integrals XII. , XIII. , XIV. , may also be written — cot -1 u + c, 
— esc -1 u + c, — covers -1 u + c, respectively. 

174. General theorems in integration. 

A, Let fix), F(x), cf>(x), • ••, denote functions of x, finite in 
number. By Arts. 29, 31, 167, the differentials of 

J [/(*) + -F(aO + 4>(as) + •• •] d« + c and 

f/(a?)efoc + (F(x)dx + ($(x)dx + ••• + c\ 

are each /(a?) dx + i^(ic) cfa; + <j>(x)dx + 

Hence, £/ie integral of the sum of a finite number of functions and 
the sum of the integrals of the several functions are the same in the 
terms depending on the variable, and can differ at most only by an 
arbitrary constant. 

(For integration of the sum of an infinite number of functions, see 
Art. 197.) 

EXAMPLES. 

1. ( (x 3 + cos x + e x )dx = \ x 3 dx + \ cos x dx + \ e x dx + c 

= |x 4 + sinx + e x + c. (1) 

Note 1. Each integral in the second member in Ex. 1 has an arbitrary 
constant of integration ; but all these constants can be combined into one. 

2. i (x 5 — sin x + sec 2 x)dx = | x 6 + cos x + tan x + c. 

B. The differentials of 

\mu dx + Co and m \u dx + C\ 

are each mudx. Hence, 

a coyistant factor can be moved from either side of the integration 
sign to the other ivithout affecting the terms of the integral which 
depend on the variable. 



174.] ELEMENTARY INTEGRALS. 295 

C. The differentials of 

( u doc + Co, in \ — dx, + c lf — \mu dx + c% 9 

•/ %) ill lf¥%/ %) 

are each. udx. Hence, 

the terms of the integral ivhich depend on the variable are not affected, 
if a, constant is introduced at the same time as a multiplier on one 
side of the integration sign and as a divisor on the other. 

Note 2. Theorems B and C are useful in simplifying integrations. 

3. (1) f:3xdx = 3 (xdx = §x 2 + c. 

(2) (^g= (V*dx = x ~ 3 + c =- — -\-c. 
w J x± J _4 + l 3 x 3 

4. I 2 sin x dx = 2 \ sin x dx = — 2 cos x -f c. 

5. ( sin 2 a; dx = \ \ 2 sin 2 x dx = \ \ sin 2 a: d(2 x) = — \ cos 2 x + c. 

Note 3. A factor involving the variable cannot be moved, or introduced, 
in the manner described in theorems B and C. Thus, \ x 2 dx = i x 3 + c ; 
but x i x dx = i x 3 + c. Also, ( x' 2 dx = | x 3 + c ; but - ( x 3 dx = i x 3 + c. 

_ r. , f sin w fcZ(cosw) .- , N , 

6. I tan w dw = I du = — I — = — log (cos w) + c 

J J cos m J cos u 

= log (sec w) + c. 

_ f . , r cos it fd(sin w) . . 

7. \ cot z< du = \ du — \ -±~. + c = log (sin u) + c. 

J J sin u J sin w © \ / 

9. Write the anti-derivatives of x 7 , 6x 72 , 2x 40 , 4x~ 19 , 5x" 14 , — , ~ b , 

3xf, x^, 6^, 2^, J-, J_, _A_. 

Vx Vx 3 7 Vx™ 



10. Write the anti-differentials of v 3 dv, 7 Vt 2 dt, — du, — — ds. 

u* Vs 3 

11. Find \ax*dx, \cy/p*dt, (iVv^dv, (rVutdw. 



296 INTEGRAL CALCULUS. [Ch. XIX. 

10 (to ( 2ds ( x5dx ( ( St ~ jO a* 
Jv' Js + 2' J 7-ofi' J±t 2 -3t + ll 

13. (efdt, (be* x dx, (±e x2 xdx, (i'dx, (\0 2x dx. 

14. \ sin 3 x dx, 4 I cos 7 x ax, 9 ( sec 2 5 x dx, f sin (x + a) dx, 

f cos (2 x + a) dx, ( sec 2 (— + -\ dx. 

15. fsec2xtan2xax, f secf xtanf xdx, f dt , f »<fa , 
J J J Vl - tf* J Vl - x 4 

r i dx r ?>x 2 dx r dv r tat c 2 ax r dt 

J VI -25x 2 ' J Vl -x e ' *^ VFT^ 2 ' Jl + * 1 ' Jl+4x 2 ' Jfv^^Ti' 

r ox r x(?x r (7x r c?x 

* xV9x 2 - I J x 2 Vx* - 1 ' ^ V6x-9x 2 ' ^ V8 x - 16 x 2 

16. | (t 2 — 4) 2 dt, \ (a* + x*) 3 ax, | e* 1 * dx, \ (cos ax + sin nx) dx. 

17. Express formula II. in words. 

175. Integration aided by substitution. Integration can often be 
facilitated by the substitution of a new variable for some function 
of the given independent variable; in other words, by changing 
the independent variable. Experience is the best guide as to 
what substitution is likely to transform the given expression into 
another that is more readily integrable. The advantage of such 
change or substitution has been made manifest in working some 
of the examples in Art. 174, e.g. Exs. 5, 6, 7, 8, etc. 

EXAMPLES. 

1. f (x + a) n dx, in which n is any constant, excepting — 1. 

Put x + a = z ; then dx = dz, and 

((x + a) n dx = (V dz = -g^i- + c = ^ + a) " +1 + c. 
J J n + 1 n + 1 

This may be integrated without explicitly changing the variable. For, since 
dx = d{x + a), f (as + g) w ax = f (x + a) n d(s + «) = ^ + g ^ W+1 + c. 

2. f(x + a)-i^ = f^^=f^+^ = iog(x + a) + c. 



174,175.] ELEMENTARY INTEGRALS. 297 



r dx 

J x V4+~ 



3x 

Put 4 + 3 x = z 2 ; then x = i(z 2 — 4), and dx = f z dz. Hence, on denoting 
the integral by 7, c ' \ r ( \ 1 \ 

J ^-4 2J U-2 0+'2 



= iio g ^ + c^|iog ^i±l;- 2 + c 

* + 2 \/4 + 3x + 2 

4. f ^ , 
J Va 2 - x' 2 

Put x = a sin 6. Then c?x = a cos 6 dd, and 



J cto _ r a cose dd _ r 
Va 2 - x 2 *^ Va 2 - a 2 sin 2 *^ 



c?0 = + c = sin- 1 - + c. 
a 



This integral may be found by another substitution. For, put x = az ; then 

dx = adz i and f — ^— = f_ _^^ = f__g^_ ■ 
J Va 2 - x 2 •> Va 2 - a 2 2 2 J VI - s 2 



= sin -1 z -f- c = sin -1 - + c. 



5. (Va?-x 2 dx. 

Put x = a sin 0. Then dx = a cos d0 ; and 



f Va 2 -x 2 dx= f Va 2 - a 2 sin 2 d > acosddd = d 2 fcos 2 0d0=^ f (l + cos20)d0 

= ^( H ^li) +c ^ (Hsin ^ C o se ) + c 

1 \ A J 2i 

= ^f sin- 1 ^ + g J a2 ~ x2 ) + c = i(a 2 sin" 1 - + x Va 2 - x 2 ) + c. 

2 \ a a > a 2 / a 

This important integral may also be obtained in other ways ; see Ex. 4, 
Art. 188, and Ex. 5, Art. 176. 



• f f u v ( Put u = a *0 Ans - - tan_1 - + c - 

./ a 2 + w 2 a a 



6 



7. f du (Put w = as.) -4ms. - sec-i ^ + c. 

■^ w Vw 2 — a 2 a ct 

8. ( dU (Putw = a«.) 4ms. vers-i-^c. 
^ V2 azt — it 2 # 

q f xdx 

J VxTT 

Put Vx+T=2. Thenx+l=z 2 , dx=2zdz, and f xdx = f£ 

^ Vx + 1 J 

= 2 f (s 2 - 1) as = f z(z 2 - 3) + c = |(x - 2) Vx+1 + c. 



: 2 -l)2zaz 
z 



298 INTEGRAL CALCULUS. [Ch. XIX. 

10. (£**-** 

J Vsin x 

Put sin x = t. Then cos x dx = dt, cos 3 x dx = cos 2 x • cos x dx = ( 1 — t 2 ) dt. 

•• 1 3 , = \ — 7- -<& = K« 3 -t s )dt = %t 3 -f£ 3 + c 

•^ vsinx ^ ^3 ^ 

= f £~ 3 (4 - £ 2 ) + c = | sin 3 x (4 - sin 2 x). 

1. ( sin 5 a; cos x dx, \ tan 3 x sec 4 x dx, \ sec 2 (4 — ? x) dx, f e -2 * dx. 

f x 2 dx f(x + l) 3 , C x — 2 , f , „.i, 
2- ( t — -7TT, \ v , y d», J - <&, \ x(x - 2) 3 dx. 

J (x + l) 3 J x^ J ^+2 •> 

3. f V(x + aYdx, (y/(m + nx)*ax, f dx , f ^L_ 

J J J VS-7x J <, 



V(4 + 5?/) 3 
i^dx. 



4. (v+ w *dx, (v- 3 *cfo, r — — — , c sin ( io 

J J J (l + x 2 )tan~ix J x 

5. ft(t-lftdt, §(a+by)%dy, j"(j» + «)*<fe, fcosfxdx. 

6. |cos 3 xdx, Jsec 4 xdx, lsin 5 xdx, j sec 2 ( — j 

„ f sin a; dx f cos x dx f sec 2 x dx T 
J 3 + 7 cos x J 9 — 2 sin x J V4 — 3 tan x ' 



dd. 
nj 



sec 2 x dx 



VlO - 3 sec 2 x 



8. f x dx — , f (a 2 - x 2 ) ^x dx, f V(a 2 + x 2 ) ■ x dx, f ^ ^ x — 
->Va 2 + x 2 ■> •> J (tf-a?)i- 

176. Integration by parts. Let w and v denote functions of a 
variable, say x\ then [Art. 32 (7)] 

d (uv) = udv -\-v du, 

whence u do = d (uv) — vdu. 

Hence, on integration of both members, 

( u dv — uv — \v du, (1) 

If an expression f(x) dx is not readily integrable, it may be 
divided into two factors, u and dv say. The application of 

formula (1) will lead to the integral J v du, and it may happen 

that this integral can easily be found. 

Note 1. The method of integrating by the application of formula (1) is 
called integration by parts. This is one of the most important of the par- 
ticular methods of integration. # 



175, 176.] ELEMENTARY INTEGRALS. 299 

EXAMPLES. 

1. Find \ xe x dx. 

Put u = x ; then civ = e x dx, 

du = dx, and v = e x . 

.\ ( xe x dx = xe x — \ e x dx = xe x — e x + c. 

2. Find i sin -1 x dx. 

Put u = sin -1 x ; then dv = dx, 

dx 

du = — i and v = x. 

VI -x 2 

dx 



;. \ sin -1 x dx = x sin -1 x — \ — — 



VI — x" 
= x sin" 1 x + Vl - x 2 + c. (See Ex. 18, Art. 175.) 

3. Find j x cos x dx. 

Put u = cos x ; then dv = x dx, 

du = — sin x dx, and v = \ x 2 . 

• .*. \ x cos x dx = J x 2 cos x + | j x 2 sin as dx. 

Here the integral in the second member is not as simple a form, from the 
point of view of integration, as the given form in the first member. Accord- 
ingly, it is necessary to try another choice of the factors u and dv. 

Put u = x ; then dv = cos x dx, 

du = dx, and v = sin x. 

.°. ( x cos x dx = x sin x — \ sin x dx = x sin x + cos x + c. 

4. Find \ X s cos x dx. 

Put u = x 3 ; then dv = cos x dx, 

du = 3 x 2 dx, and « = sin x. 

.*. \ x 3 cos x dx = x 3 sin x — 3 ( x 2 sin x dx. (1) 

It is now necessary to find I x 2 sinx dx. 

Put ?« = x 2 ; then dv = sin x dx, 

dw = 2 x dx, and v = — cos x. 

.•. ( x 2 sin x dx = — x 2 cos x + 2 I x cos x dx. (2) 



300 INTEGRAL CALCULUS. [Ch. XIX. 

It is now necessary to find i x cos x dx. 

By Ex. 3, ( x cos x dx = x sin x + cos x -}- c. 

Substitution of this result in (2), and then substitution of result (2) in 
(1), gives 

( x s cos x dx = x 3 sin x + 3 x 2 cos x — 6 x sin x — 6 cos x + C\. 

When tne operation of integrating by parts has to be performed several 
times in succession, weakness in arranging work is a great aid in preventing 
mistakes. The work above may be arranged much more neatly; thus: 

\ x 3 cos x dx = x s sin x — 3 \ x 2 sin x dx 

= x 3 sin x — 3 — x 1 cos x + 2 j x cos x dx 

= x 3 since — 3[— x 2 cosx + 2(x sinx + cosx + c)] 
= x 3 sin x + 3 x 2 cos x — 6 x sin x — 6 cos x + (7 
=: x(x 2 — 6) sin x + 3(x 2 — 2) cosx + C 

The subsidiary work may be kept in another place. 

• 
5. Find (* Va 2 - x 2 dx. (See Ex. 5, Art. 175.) 



Put u = Va 2 — x 2 ; then dv = dx, 

xdx 

Va 2 - x 2 



du _ xdx ? and 



.-. f Va 2 - x 2 dx = xVa 2 - x 2 + f x * dx • (1) 

J J Va 2 - x 2 



Now Va 2 — x 2 = 



a* - x< a' 



Va 2 - x 2 Va 2 - x 2 Va 2 - x 2 

hence x * = ^ -Va 2 -x 2 . 

Va 2 - x 2 Va 2 - x 2 

Substitution in (1) gives 

CVa 2 -x 2 dx = xVa 2 -x 2 + ( ^ dx - (Va 2 -x 2 dx, (2) 

J J Va 2 - x 2 J 

Hence, on transposition of the last integral in (2) to the first member, 
division by 2, and Ex. 4, Art. 175, 

f Va 2 - x 2 dx = -(x \/a 2 - x 2 f a 2 sin" 1 -Y 



176, 177.] ELEMENTARY INTEGRALS. 301 

6. ( e x cos x dx = \ e x (sin x + cos x). 

(Integrate, putting u = e x ; then integrate, putting u = cos as. Take half 
the sum of the two results.) 

7. \ xe ax dx. 11. (xlogxdx. 15. | x 2 siu x c?x. 

8. \xe~ x dx. 12. (x 2 logxdx. 16. \ e x x m dx. 

9. ix 2 e a dx. 13. Jtan _1 xdx. 17. I x sin x cos x dx. 
10. flogxtfx. 14. fx tan- 1 a; da;. 18. f- sm ~ lx dx. 

J J J y 1 _ X 2 

19. Derive I e x sin x dx = \ e x (sin x — cos x). (See Ex. 6.) 



177. Further elementary integrals. A further list of elementary 
integrals is given here. They can be verified by differentiation. 
Some of the ways in which, they may be derived are indicated in 
the latter part of the article. 

XY. f tan u du = log sec u + c. 
XVI. j cot u du = log sin u + c, 

XVII. ( sec udu — log (sec u + tais u) + c 9 

= logtan(| + |) + c. 

XVIII. I cosec u du = log tan ^ + c, 

XIX. f du =sin-^+c. 
•> V a 2 _ u 2 a 

XX. f-J^_ = lten-i^+c. 

J a 2 + u 2 a a 

XXI. C gg ^Jsec-^ + c. 

XXII. f ^ =Ters-i^+c. 
N.B. See Note 1. 



302 INTEGRAL CALCULUS. [Cii. XIX. 

XXIV. f dM = log (U + V^2 + a 2) + C9 

J v'u 2 + a 2 

a 
du 



XXY. f CT " = log Cm + VW 2 - a 2 ) + c, 

J ^u 2 - a 1 

a 

XXVI. f V a 2 - «*2 du = l(u V a * - u 2 + a 2 sin 1 - 

Integral XXII. is also reducible to form XIX. For 2 au — u 2 

= a 2 — (w — a) 2 , and dw = d (w — a) ; 

/# f du = r d(u-a) ^ dir i«-« +< ,. 
J V2 at* - t* 2 J Va 2 - (ti - a) 2 a 

Ex. Show that this result and that in XXII. are equivalent. 
Remarks on integrals XV. to XXVI. 

Formulas XV., XVI. For derivation, see Exs. 6, 7, Art. 174. 
Formulas XVII., XVIII. 

cosec u — cot u 



Since cosec u = cosec u 



cosec u — cot u 



J„~™„ „. ,7 f — cosec m cot 2t + cosec 2 u ,„ 
cosec w aw = I — aw 
J cosec w — cot u 

= fd (cosec w- cot w) = lQg (cogec u _ CQt M) 
J cosec w — cot w 



l^„l — COS U i 



Substitution of u + — for ^f in the last two lines gives 
(cosec iu-\--\du = log tan (- + -), i.e. (sec u du = log tan (r + j) > 

= log j cosec ( w+-j— cot | m+- j ^ = log(secw+tanw). 
There are various methods of deriving XVII. and XVIII. 



177.] ELEMENTARY INTEGRALS. 303 

Formulas XIX., XX., XXI., XXII., XXIII. For derivation, 
see Exs. 4, 6, 7, 8, Art. 175, and the following suggestion : 

Suggestion: — ; = — ( ; — J I -5 o = ^~ ( — ; 1 V 

u 2 — a 2 2a\u — a u + aj a z — u z 2a\a + u a — uj 

Formula XXIV. 

Put u 2 + a 2 = z 2 ; then u du = z dz, whence — = — • 

z u 

„ da du dz 

Hence, ■ = — = — 

Vu 2 + a 2 z u 

~ . A . du du + dz d (u + z) 
On composition, — === = — — = — * — ! — J — 

\/u 2 + a 2 u + z u + z 

... C du = C d (u + z) = lQg , + ^ + c = i g ( w + Vw2 -1- «2) +c . 
J Vw 2 + a 2 J u + s 

The last result may be written 

u + Vu 2 + a 2 



log (m + Vw 2 + a 2 ) - log a + c 1 , i. e. log " "*" v "*" " + c', 
a form which is convenient for some purposes. See Note 3. 

Formula XXV. can be derived in the same way as XXIV. 

Formula XXVI. For derivation, see Ex. 5, Art. 175, and Ex. 
5, Art. 176. 

Note 1. Integrals XIX., XX., XXI., XXII., may be respectively written 

- cos" 1 - + c', - - cot" 1 - + c', - - esc" 1 - + c', - covers" 1 - + c'. 
a a a a a a 

Ex. Show this. 

Note 2. Integrals XXIII., XXIV., XXV., may be written thus : 

f -^L_ = 1 h y tan-i « + c'(i< 2 < a 2 ), 
J w 2 - a 2 a a 





u z — a* 


a 


a 


r du 

^ V M 2 + a 2 


= hysin _1 


a 


J: 


flu 


= ± hy cos 


-l^ + C. 
a 



304 INTEGRAL CALCULUS. [Ch. XIX. 

The functions whose symbols are here indicated are the inverse hyperbolic 

tangent of -, the inverse hyperbolic sine of -, and the inverse hyperbolic 

u a a 

cosine of _. For a note on hyperbolic functions see Appendix, Note A. 

The close similarity between XX. and these forms of XXIII. may be remarked ; 
so also, between the forms of XIX. and these forms of XXIV. and XXV. 

Note 3. The same integral may be obtained by various substitutions, and 
may be expressed in a variety of forms. Instances of this have already been 
given; another example is the following : Integral XXIV. can also be derived 
by changing the variable from u to z by means of the substitution Vu 2 + a 2 
= z — u ; this leads to the form 



J 



= log (« + Vu 2 + a 2 ) + c. 



V u 2 + 



The first member can also be integrated by changing the integral from u 
to z by means of the substitution Vu 2 + a 2 = zu ; this leads to the form 

f du = log (Vg+Z±^|^ +C , 

J Vu 2 + a 2 *- Vu 2 + a 2 - u > 

It is left as an exercise for the student, to employ these substitutions in 
the integration of XXIV., and, the arbitrary constants of integration being 
excepted, to show the identity of the various forms obtained for the integral. 



EXAMPLES. 

1. f 4 + 7 x dx = ((— — + _IiL_Y?x = 2tan-i^ + -log(4 + z 2 )+c. 
J4 + x 2 J ^4-f x2^4 + x 2 J 2 2 aK J 

2. f 4 + 7a: dx= (7 4 + 7X )dx=4sm-i*-7(4-x 2 ) 2 +c. 
J Vi-x 2 J VV4-^ V4-x 2 / 2 

3. f ** = f <g(» + 2) = l tan -i^±2 

Jx 2 + 4x + 20 J(x + 2) 2 + 16 4 4 

4a. f g _ = f — g_ (a; + 2) = lo ^ x+2 4-Vx 2 +4x+2(r)4-c. 

J Vx 2 + 4x + 20 J V(x + 2) 2 + 10 

4 6. f dx =f ^ + 2 ^ ^rin-ig+J + c- 

J Vl2 - x 2 - 4 x J Vl6 - (x 4- 2) 2 4 

Notice should be taken of the aid afforded (e.g. in Exs. 3, 4 a, 4 6) by 
completing a square involving the terms in x. 

S. f k = = 1 f <^ 2 *> = iseo-i V* + c. 

Jlx^/i x 2_g 7 J2xV(2x) 2 -3 2 21 3 



177, 178.] ELEMENTARY INTEGRALS. 305 



6. 



Jx 2 



dx 

1 r™_ *- 



*Vl6 - x 2 



Put x = -- Then dx = dt, and 

« £ 2 



7. 

8. 

9. 

10. 

11. 

12. 

13. 
14. 



r dx r — y^__ = _i f(i6t 2 -i)"^(Z(i6^_i) 

J *Vl6 - x 2 •* Vl6 *- - 1 82 J l 

=-^ w '- 1 >* + *=- fl2 s? I ' + . ft 

Jx 2 + 6x + 17' " J Vl7 + 6x-x 2 ' * Vx 2 + 6 x + 10 

i) f fe ( 2 ) r g ; (3) r___^___. 

; J7-6x-x 2 ' JV7-5x-x 2 ' JVx 2 -5x + 7 

1) f *5 ; (2) f *> ; (3) f dx , ■ 

'Jx 2 + 5x-2' w Jx 2 + 5x-9 J V4x 2 -3x + 5 



n C (to f2) C ^ C3) f dx 

y J4x 2 -5x-f6' v y J V9-5x-4x 2 ' J7-5x-4x 2 

1) f ^ x ; (2) f ^ x ; (3) f cfa 

•^ Vrf x — x 2 *^ V9 x — 4 x 2 J 5 xV9x 2 — i 



25 

_2 C?X. 



1) f dx ; (2) f v / 9^ 2 dx; (3) f V25^ 

J (x-l)Vx 2 -2x-3 J j0 

1) f V36 - 4 x 2 dx ; (2) fsec3xdx; (3) f cosec (4 x — a) dx. 

1) (*tan(3x + a)dx; (2) fcot (4x 2 + a 2 )xdx ; (3) fsec2xdx. 



15. Derive integrals 62 a, 6, 63 a, 6, p. 406. 



, V25 — x 2 j_ f dx f dx 






(4 _|_ x 2 ) 2 xvl2 x — x 2 

178. Integration of f(x)dx when f{x) is a rational fraction. 

In order to find \f(x)dx when f(x) is a rational fraction, the 

procedure is as follows : 

Resolve f(x) into component fractions, and integrate the differ- 
entials involving the component fractions. 

Xote. It is here taken for granted that in his course in algebra' the 
student has been made familiar with the decomposition of a rational fraction 
into component fractions, or, as it is usually termed, the resolution of a 
rational fraction into partial fractions. Reference may be made to works 
on algebra, e.g. Chrystal, Algebra, Part I., Chap. VIII. ; also to texts on 
calculus, e.g. Snyder and Hutchinson, Calculus, Arts. 132-137. 



B06 INTEGBAL CALCULUS. [Ch. XIX. 

Examples 1, 2, 4 will serve to recall to mind the practical 
points that are necessary for present purposes. 



,p 



EXAMPLES. 

3x 2 + 4x+J4^ 



X 2 + X 

Here x*-Sx* + 4x + U = x _± + 14a -10 



x 2 + x - (3 x 2 + a; — 6 

The fraction in the second member is a proper fraction, and is iw fts 
ZoioesZ terms. Accordingly, the work of resolving it into fractions having 
denominators of lower degree than the second, may be proceeded with. 
Since its denominator, x 2 + x - 6, i.e. (x — 2) (x + 3), is the common denom- 
inator of the component fractions, one of the latter evidently must have a 
denominator x — 2, and the other a denominator x + 3. Since these frac- 
tions must be proper fractions, their numerators must be of lower degrees 
than the denominators, and, accordingly, must be constants. 
Accordingly, put 

14x-10 / 14 a: -10 \_ A B m 

x 2 + x - 6 \ (x - 2) (x + 3) / x - 2 x + 3 ^ ) 

Here A and B are to be determined so that the two members of (1) shall 
be identically equal. 

On clearing of fractions, 

14x-10 = ^(x + 3)+^(x-2). (2) 

Since the members of (2) are to be identically equal, the coefficients of 
like powers of x must be equal. That is, 

A + B = 14, 
3 A -2 B= -10. 

On solving these equations, A = ^, B = - 5 g 2 . 

. fx3_ 3 ,- 2 + 4x+14 dx= Cf _ 4+ 18 52 \ 

J x* + x-6 JV 5(x - 2) 5(x + 3) I 

= ^ X 2_ 4:X+ i^iog ( X - 2)+ 4* log (x + 3)+ c. 
Another way of finding A and B in (2) is the following : 
The two members of (2) are to be identically equal, and accordingly equal 

for all values of x. 

Now, put x = — 3 ; then — 5 B = — 52 ; whence, B = - 5 j 2 -. 
Put x = 2 ; then 5 A = 18 ; whence, ^1 - - 1 /- 

Note 1. Any other values, e.g. 3 and 7, may be assigned to x ; in this 

case, however, the values 2 and — 3 give the most convenient equations for 

determining A and B. 

Note 2. For a more rapid way of finding A and B in such cases as (1), 

see Murray, Integral Calculus, Appendix, Note A. 



178.] ELEMENTARY INTEGRALS. 307 






r + « a - o a; + d 

The fraction in the integrand is a proper fraction, and is in its lowest 
terms. Accordingly, the work of decomposing it into fractions having de- 
nominators of degrees lower than the third may be proceeded with. Since 
the denominator x 3 + x 2 — 5 x + 3, i.e. (x — l) 2 (x + 3) is the common 
denominator of the component fractions, one of the latter evidently must 
have a denominator x + 3, and another must have a denominator (x — J) 2 . 
It is also possible that there may be a component fraction having the denom- 
inator x— 1; for, if there is such a fraction, it does not affect the given 
common denominator. Accordingly, put 

x 2 + 21 x - 10 A . B , C /QN 

+ 71 TTo + Z 7' W 



(x - l) 2 (x + 3) x + 3 (x - l) 2 x - 1 

in which A, B, C are constants to be determined. 

On clearing of fractions, equating like powers of x (for reasons indicated 
in Ex. 1), and solving for A, B, C, it is found that 

A = - 4, B = 3, = 5. 

f x 2 + 21x-10 dx= n^± + 3 + _6_\ (fa 
J x 3 + x 2 - 5 x + 3 J \x + 3 (x - l) 2 x-lJ 

= 51og(x-l)-41og(x + 3)--^-+c = log<^4r?-^-r+ c - 

x — 1 (x -f 3) 4 x - 1 

Note 3. It may be asked why the numerator assigned to the quadratic 
denominator (x — l) 2 in the second member of (3) is not an expression of 
the first degree in x, say Bx + D, instead of a constant. The reason is, that 
if such a numerator were assigned, the fraction would immediately reduce to 
the forms in (3) . For 

Bx + D _ £(x-l)+ Z) + B _ B D + B , 

(x-1) 2 (x-1) 2 x-1 (x-1) 2 ' 

forms which appear in (3). 

Note 4. If a factor of the form (x — a) r appears among the factors of the 
denominator of the fraction to be resolved, there evidently must be a com- 
ponent fraction having (x — a) r for its denominator. There may also possi- 
bly be fractions having as denominators (x — a) of various powers less than 
r, e.g. (x — a) r ~\ (x — a)'' -2 , •••, x — a. Accordingly, in such a case it is 
necessary to allow also for the possibility of the existence of fractions of the 

forms 

M F L 

(x — a) r_1 ' (x — a) r ~ 2 ' x — a 

in which M, F, •••, L, are constants. 



308 INTEGRAL CALCULUS. [Ch. XIX. 

J 2 x 2 — 8 a; — 10 
— dx. (Compare denominators in Exs. 2, 3.) 
x 4* x — 5 x 4~ o 



J; 



5x 2 + 3x+17 ^ 
c 3 — x 2 + 4 x — 4 



The fraction in the integrand is a proper fraction and is in its lowest terms. 
If it were not so, division as in Ex. 1 and reduction would be necessary. 
Since the denominator x 3 — x 2 4- 4 x — 4, i.e. (x 2 + 4) (x — 1) , is the com- 
mon denominator of the component fractions, one of the latter must have a 
denominator x 2 + 4, and the other a denominator x — 1. Accordingly, put 

5 x 2 4- 3 x + 17 _ .4x + B , C 



(x 2 + 4) (x - 1) x 2 + 4 

in which A, Z?, C, are constants to be determined. 

On clearing of fractions, equating coefficients of like powers of x, and 
solving for A, J5, C, it is found that 

A = 0, B = 3, = 5. 

. r5x; 2 + 3x + i7 dx= Cf_s_ + _ L _\ dx 

Jx 3 -x 2 + 4x-4 JU 2 + 4 x-l) 

- - tan- 1 - + 5 log (x - 1) + c. 

2 2 

Note 5. The expression x 2 + 4 has factors x + 2 i, x — 2 i (i = V— 1) ; 
if these be taken, component fractions imaginary in form, are obtained. It 
is usual, however, not to carry the decomposition of a fraction as far as the 
stage in which component fractions imaginary in form may appear. 

Note 6. The numerator Ax -}- B is assigned above ; for the numerator 
over a quadratic denominator whose factors are imaginary, may have the 
form of the most general expression of the first degree in x. 

Note 7. When a quadratic expression x 2 + px + Q has imaginary factors 
and is repeated r times in the denominator of a fraction, in the process of 
decomposition of this fraction allowance must be made for fractions of the 

forms, Ax + B Cx + D _ Mx + N . 

(x 2 +px + g) r ' (x 2 +px + g) r_1 ' ' x 2 -fj5x + g 

5. (1) (-11^-4x4-28^ f 3x 2 -13x-5 

denominators in Exs. 4, 5.) 



178, 179.] ELEMENTARY INTEGRALS. 309 

Find the anti-derivatives of the following fractions : 

6 a + 37 n 2x 2 

x 2 -3x-28* 

7 . ZX+ 1 _., 18. 

19. 
20. 

10. — _.. 21. 

x(2x 2 + 3x- 5) 

11. ; g+jL r . 22. 

12. 

x*-13x 2 + 36 x 3 + 3x 

13 gLLJ 24 2 x 3 - x 2 + 8 x + 12 
' (x-1) 2 ' ' x 2 (x 2 + 4) 

14 8x + 5 g5 2 + 3x-x 2 



x + 37 




x 2 - 3 x - 28 




8x+ 1 




2x 2 -9x— 35 




X 3 _ 2 X 2 - 1 




x 2 -l 




X 4 - X 2 + 1 




X 3 — X 




x 2 - 10 x - 5 




x(2x 2 + 3x- 5) 




x 2 + pq 




x(x — p)(x + q) 




llx 3 -llx 2 -74x 


+ 84 



(x + 1) 3 


x 2 - 3 x + 3 


x(x 2 + 3) 


12 - x - x 2 


(3x-2)(x 2 + 5) 


(x+l) 2 _ 


X 3 + X 


x 3 -l 


x 3 + 3x 


2 x 2 + 3 x + 6 


x 3 + 3x 


7x 2 + 9 



(4x+5) 2 (x-l)(x 2 -2x+5) 

5 x 2 + x - 10 1 „ 6 1 + 7 x + x- + x 3 
x 2 (2x + 5) ' (x 2 + l) a 

30x 2 + 43x -8 



(x + 4)(3x + 2) 2 

Ex. 27. Show that any expression of the form C ( mx+ glgg i n which 

«/ nifi -4- 7)T -4- f* 
m, w, a, 6, and c are constants, is integrable. 

179. Integration of a total differential. In Art. 85 it has been 
shown that the necessary condition for the existence of a function 

having POx+Qdy (1) 

for its differential, is that —■ = |& (2) 

It has also been stated (Art. 85, Note 1) that condition (2) is 
sufficient for the existence of such a function. In other words, 
if the expression (1) has an anti-differential (or integral), relation 
(2) must be satisfied; conversely, if relation (2) is satisfied, the 
expression (1) has an integral. Accordingly, relation (2) is called 
the criterion of integrability for the expression (1). If this criterion 



310 INTEGRAL CALCULUS. [Ch. XIX. 

is satisfied, the expression (1) is said to be a complete differential, 

a total differential, and also an exact differential. 

If test (2) is satisfied, the integral of (1) can easily be found. 

This integral's partial cc-differential, Pdx, can only come from 

terms containing x (Art. 79). Hence, the integral of Pdx with 

respect to x, namely, f 

\Pdx + c, (3) 

must yield all the terms of the required integral that contain x. 
Also, Qdy can only come from terms containing y. Hence the 
integral of Q dy with respect to y, namely, 



/ 



Qdy + c 2 (4) 



must yield all the terms of the required integral that contain y. 
Some of these terms may contain x\ if so, they have already been 
obtained in (3), and need not be taken this second time. Hence, 
if the integral of a differential of the form 

Pdx+ Qdy 

is required, apply the test for integy -ability, namely, 

dP^dQ. 

dy dx ' 

if this test is satisfied, integrate Pdx ivith respect to x ; then integrate 
Qdy with respect to y, neglecting terms already obtained in I Pdx ; 
add the results and the arbitrary constant of integration. 

EXAMPLES. 

1. Integrate (2 xy + 2 + 3 y 2 + 12 x) dx + (x 2 + 6 xy + 4 ?/ 3 ) dy. 
Here P = 2 xy + 2 + 3 y 2 + 12 x, and Q = x 2 + 6 xy + 4 y\ 

.-. <^=2z + 6y, and^ = 2x+6y. 

dy dx 

Thus the criterion of integrability is satisfied. 
Also f Pdx = x 2 y + 2 x + 3 xy 2 + 6 x 2 ; 

and \ Q dy = x 2 y + 3 xy 2 4- y 4 , in which y* has not been already obtained 
in ( Pdx. Hence the integral is 

x 2 y + 2 x + 3 y 2 + 6 x 2 + y 4 + c. 



179.] ELEMENTARY INTEGRALS. 311 

2. Verify the result in Ex. 1 by differentiation. 

3. Find ( (x dy — y dx). 

Here ^M — 1, and — = — 1 ; hence the test for integrability is not satis- 
dx dy 

fled, and there is not an anti-differential. 

4. (1) (e x (cosy dx- siny dy). (2) f [(3x 2 4-8x?/4-4)dx4-(4x 2 -6)d?/]. 

5. Integrate: (1) cos x sec 2 y dy — (sin x tan y + cos x) dx. 

(2) (xey - 2 x) dy + (e» - 2 y + 2 x) dx. (3) (3 - 4 x - y) dx - (x 4- y) dy. 

N.B. An accurate and ready memory of the fundamental inte- 
grals (Arts. 173, 177), resourcefulness in making substitutions 
(Art. 175), and quickness in integrating by parts (Art. 176), are 
three very important things to cultivate in order to insure com- 
fortable progress in the study of the calculus. 

EXAMPLES. 

1. (ln 2 x^+ m dx, f (a + 6)x 2 («+ 6 )-!dx, f(r + s)z n +*+' 2 dz, ( rh%y r *-*dy, 

Jo t + 2 J v 2 + 3 J x 2 - 2 J 9 1* + 20 

f_&L_ f (\aUyhdy, f^ , f *** , f g2 ~ 1 dz. 
J 2 2 - 12 Ji v . y *' J -v/iT=~x6 J Vx^Tq J (2 z - l) 2 

2. f tan (mx + n) dx, f (sec 3 x + 2) 2 dx, f tan 2 d0, f ''sin I- + -") d0. 

6 

3. |cos -1 xdx, jsec _1 xdx, Jcot _1 xdx, |(logx) 2 dx, ( x 2 e a dx, 
( x s e~ x dx, \ sin x log cos x, I x w log x. 

4. fsLi*,, f_i«L,b, r»_*_^ fJiUfc 

Jo 2 Jo e 3x Jo Ji Vn^x 2 

6 f <?. sin0d0 f (1 4- cos0) d0 f dx f secxtanxdx 

J w + n cos (9 J sin J sin x + cos x J (tan 2 x — 3) 2 ' 

r do ( ri og 2 (fflg + w) ^ r dx r_dx_ 

^ cos 2 0V4- tan 2 0' J «*« + w ' J Va 2x - m 2 ' J e* + e -x ' 



312 INTEGRAL CALCULUS. [Ch. XIX. 

sin - dx 

c dx r de r 2 

J e 2x _ e -2x J C0S 2 2 _ S i n 2 2 d J . X I X 

sin - -v cos - 
4 \ 2 

j [ (1 — sin x cos y) dx — (cos x sin ?/ + 2 y) dy] , 
J [(1 — sin x sin y) dx -f (cos x cosy — 1) dy]. 
7. Derive the following integrals : 

J 2(n + l) JVx 2 ±a 2 



= -Va 2 



(3) f x(a 2 _ x 2)»dx - - ( f "f^ W f -p_ 

J 2(w + 1) J Va 2 - x 2 

8. Derive the following integrals : 

(1) f^ = Iiog(« + & x ). (2) C( a + bx)-dx = ^±^^, whenn 
./ a + 5x 5 J &(w 4- 1) 

is different from - 1. (3) C xdx = 1 [ a + bx - a log (a + &x)]. 

J a + bx b 2 

(4) r_«^ = i [4(a + 6a . )2 __ 2a(a + 6a . ) + a 2 log(a + 6a . )]e(6 ) r g« 

J a + bx b 2, J x(a + &x) 

== -llog a + te . ( 6 ) f ** = _l + Alog«+^. (7) f xdx 

a x w J x 2 (a + bx) ax a* x v J (a + fox) 2 

= I[lo g (« + 6a: ) + -^-]. 

9. Derive the following integrals : 

(1) JV^fV^^r W f-rh = -7= tan_lje V-- when 

J a 2 — b-x 2 2 ab a — bx J a + fox 2 y^ * a 

a>0and&>0. (3) (* *<** = A i og ( x 2 + g\ . ( 4 ) f x 2 dx = x_ 
w Ja + 6aJ a 2 b a \ b) KJ Ja+btf b 

gf da? • (5) f ^ = J_log_^_. (0) f <** 

&Ja + 6x 2 v y J x(a + &x 2 ) 2 a a + bx 2 J J x 2 (a 

_b C dx ,„. C xdx 

aJa + bx 2 J (a+bx 1 )" 2 b(n -l)(a + &X 2 )"- 1 



(a + 6x 2 ) 2 a a + fox 2 J x 2 (a + &x 2 ) ax 

1 



10. Derive the following integrals 



(1) §xVg-TVxdx =- 2 ( 2a ~ 3 6x)V(a + &*) 3 , (2 ) J^V^T^ <fe = 

2(8a 2 -12a&x+15fi 2 x 2 )V(a+&x) 3 ^ f xdx _ 2(2 a- &x) ^ ^ 

105 63 ; J VaT&x" 3 6 2 

(4) f x 2 dx = 2(8a 2 --4a5x + 3& 2 x 2 ) Vcr p^_ (5) j' dx = 

J Va + &x !5& 3 ^Va + fcx 

J_ ]og Va + to-^ for a > . _2_ to -x j«_+te for a < 0. 
Va Va + bx + Va V— a — a 



CHAPTER XX. 

SIMPLE GEOMETRICAL APPLICATIONS OF 
INTEGRATION. 

180. This chapter treats of some simple geometrical applica- 
tions of integration. Examples of some of these applications 
have already appeared in Arts. 166, 167. In Art. 181 integra- 
tion is used in measuring plane areas, in Art. 182 in measuring 
the volumes of solids of revolution. In Art. 183 the equations 
of curves are deduced from given properties whose expression 
involves derivatives or differentials. 

N.B. The student is strongly recommended to draw the figure for each 
example. In the case of examples which are solved in the text he will find 
it extremely beneficial to solve, or try to solve, the examples independently 
of the book. 

181. Areas of curves : Cartesian coordinates. 

A. Rectangular axes. In Art. 166 it has been shown that for 
a figure bounded by the curve 

the a>axis, and the two ordinates for which x = a and x = b respec- 
tively, the axes being rectangular, area of figure = limit of sum of 
quantities y A x (or f(x) Ax) when Ax approaches zero and x varies 

continuously from a to b. This limit is denoted by j y dx or 

fix) dx ; it is obtained by finding the anti-differential of fix) dx, 

substituting b and a in turn for x in this anti-differential, and 
taking the difference between the results of the substitutions. 
In fewer words : the number of units in the area is the same as the 
number of units in a certain definite integral; namely, 

area of figure = ( y dx — \ /(as) dx, (1) 

Ja Jft 

The infinitesimal differential y dx is called an element of area. 

••513 



314 



INTEGRAL CALCULUS. 



[Ch. XX. 



N.B. It will be found that in many problems it is necessaiy : 

(1) To find a differential expression for an infinitesimal element of area, 
or volume, or length, etc. , as the case may be. 

(2) To reduce this expression to another involving only a single variable. 

(3) To integrate the second expression between limits (end- values of the 
variable), which are either assigned or determinable. 

B. Oblique axes. Suppose that the axes are inclined at an 
angle w, and that the area of the 
figure bounded by the curve whose 
equation is y=f(x), the #-axis, and 
the ordinates AP and BQ (for which 
x = a and x = b respectively), is 
required. Let RM be a parallelo- 
gram inscribed between A and B, as 
rectangles were inscribed in the 
figures in Arts. 165, 166. 

Area of PM = yAx • sin w. 

Area APQB = limit of sum of all the parallelograms like 
RM, infinite in number, that can be inscribed between AP and 
BQ ; that is, 




area 



Xx=b > r*h 

y sin o> • dx = sin « I y dx. 



Unless otherwise specified, the axes used in the examples in 
this chapter are rectangular. 



EXAMPLES. 

Find the area between the line 2y— 5x — 7 = 0, the sc-axis, and the 
ordinates for which x = 2 and x = 5. 

The rectangle PM represents an element of area, y dx. 
The area required is the limit of the sum of these element- 
ary rectangles, infinite in number, from AB to DC. 
That is, 

= 36| square units. 

If the unit of length used in drawing the figure 
were one inch, the figure would contain 36| square 
Fig. 105. inches. 




area 



181.] 



ABE AS OF CURVES. 



315 



2. Solve Ex. 1 without the calculus, and thus verify the result obtained by 

the calculus. 



p u,y) 

L 




Fig. 106. 



3. (a) Find the area of the circle 
x 2 + y 2 = 9 ; (b) find the area of the figure 
bounded by this circle, and the chords for 
which x — 1 and x = 2. 

Let APB be the circle whose equation 
is x 2 + y 2 = 9. Take a rectangle PM, sup- 
posed to be infinitesimal, with a width dx, 
for the element of area. Its area is ydx. 
The area of the quadrant AOB is the limit 
of the sum of all these elements of area, 
infinite in number, between O and A. 
Hence, 



OAB = (^ydx = f 3 V9 - x 2 dx = | He V9 - a? + 9 sin- 1 !!^ |tt sq. units. 

.*. area circle = 4 ■ OAB = 9ir square units. 

(6) Draw the ordinates TB and NL at the points T and N where x = 1 
and x = 2 respectively. The area of TRLXis equal to the limit of the sum 
of all the elements of area, PM, that lie between TB and NL. That is, 

area TBLX=(^ 2 ydx = CV9 - x 2 dx = J |~a:V9 - z 2 + 9 sin-^] 2 

= i{(2 V5 + 9 sin-if) - ( V8 + 9 sin-ii)} 

= VE— V2 + | (sin- 1 !- sin-ii). 

Here the radian measures of the angles are to be employed. 
Now 

V2 = 1.414 ; sin- 1 ! = (41° 40.8') = .727 radians ; sin- 1 ! = .340 radians. 

.•. area required = 2 • TBLX= 5.126 square units. 

Note 1. Other end- values of x may be used in finding the area of this 
circle. Thus 

area circle = 2^.5.4 =2 f 3 ydx = 2 f 3 V9-x 2 dx = ^xV9-x 2 + 9sin- 1 -T 
= 9 sin-n - 9 sin- 1 (- 1) =^Z-9(-- > ) = 97r square units. 

Note 2. These problems may be stated thus : Find by the calculus (a) the 
area of a circle of radius 3, (b) the area of a segment between two parallel 
chords, distant 1 and 2 units, respectively, from the centre. In this case it 
is necessary to choose axes (as conveniently as possible), to find the equation 
of the circle, and then to proceed as above. 



316 



INTEGRAL CALCULUS. 



[Ch. XX, 



4. Find the area between the curve y = 2 x 3 , the y-axis, and the lines 
y = 2 and ?/ = 4. 

The area is represented by ABLE. At any point 
P(x, y) on the arc EL take for the element of area an 
infinitesimal rectangle MP. Its area is x dy. 



ry=4 1 /M 1 

.*. area AELB— \ xdy = — I ?/ 3 dy 

23 L* _n 2 3 * 

= |~. 2* (2*-l) =1(^16-1) = 2.2797. 

'4 2¥ 2 




Fig. 107 



Note 3. The definite integral which gives the area may also be expressed 
in terms of x. For, since y = 2x 3 , dy = 6x 2 dx; also, when y = 2, as=l, 
and when y = 4, x = \/2. 



•. area .4i?Z£ = P 4 x cfy = (*_ 6 x 3 c?x = § (S/IU - 1) = 2. 



2797. 



5. (a) Find the area of the figure bounded by the parabola y 2 = 4 ax, 
the x-axis, and the ordinate for which x — X\. Show that this area is equal 
to two-thirds of the rectangle circumscribing the figure. (6) Find the area 
bounded by the parabola y 2 = 9 x, and the chords for which x = 4 and 
x = 9. 

6. Find the area between the curve y 2 = 4 x, the axis of y, and the line 
whose equation is y = 6. 

7. Find the area included between the parabolas whose equations are 

y 2 = 8 x and x 2 = 8 y. 
4* The parabolas are OML and Oi?£ ; the area of 

gf OELMO is required. To find the points of inter- 

ji^J® section of the curve, solve these equations simul- 
taneously. This gives (0, 0) the point 0, which 
is otherwise apparent, and (8, 8) the point L. 

Area OELMO = area OELN - area OMLN 
= V8 Cx^dx-l Cx 2 dx 
= H ~ _ "¥ = 2 H square units. 




8. Find the area included between the parabolas whose equations are 
3 y 2 = 25 x and 5 x 2 = 9 y. 



181.] 



AREAS OF CURVES. 



317 



9. Find the area included between the parabola (y — x — 3)'' = x, the 
axes of coordinates, and the line x = 9. Figure 52 shows that this problem 
is ambiguous, for OTGML and OTKNL are each 
bounded as described. On solving the equation of 
the curve for y, 

y = x ± Vx + 3. 

Thus if OQ = x, QG = x + Vx + 3, 
and QK = x — Vx + S. 

.-. area OTGML 

= I (x + Vx + 3) dx = 85^ square units ; 
and area OTKNL 




= \ (x — Vx + 3) dx = 49| square units. 



Also, the area MTN (the figure bounded by the 
curve and the chord for which x = 9) = area OTGML — area OTKNL 
= 36 square units, 

The area of ilf TJVcan also be found as follows : 

Area MTN = limit of sum of infinite number of infinitesimal strips, like 
KG, lying between T and MN. 

Now strip KG = (QG - QK) dx = 2Vx dx. 

.-. area MTN= (\ Vx dx = 36. 

10. Apply the second method used in finding area MTN in Ex. 9 to find- 
ing the areas in Exs. 7 and 8. 

11. Find in two ways the area between the parabola (y — x — 5) 2 = x and 
the chord for which x = 5. 

12. Find the area between the parabola y = x 2 — 8 x 
+ 12, the x-axis, and the ordinates at x = 1 and x = 9. 

Area = i ?/ dx = I (x 2 — 8 x + 12) dx 

= 18| square units. (1) 

The parabola crosses the x-axis at B and C where 

x = 2 and x = 6. 

Area APB = (*^*y dx = 2±; 

area ££C = f y dx = - lOf ; 

area COD = f w dx = 27. 

' J6 y Fig. 110. 




318 



INTEGRAL CALCULUS. 



[Ch. XX. 



Area required = area APB -f area BEC + area CQD 
= 21 _ 10| + 27 = 18$, as in (1). 

The sign of the area BEC comes out negative, because the element of area, 
y dx, is negative as x increases from OB to OC ; for dx is then positive and y 
is negative. On the other hand as x proceeds from A to B and from C to D, 
y dx is positive. The actual area shaded in the figure is 2^ + lOf + 27, i.e. 
40 square units. 

N.B. It should be carefully observed, as illustrated in this example, that 
in the calculus method of finding areas bounded by a curve, the x-axis, and 
a pair of ordinates, areas above the x-axis come out with a positive, and areas 
below the x-axis come out with a negative sign. Accordingly, the calculus 
gives the algebraic sum of these areas ; and this is really the difference between 
the areas above the x-axis and the areas below it. 

13. (a) Find the area bounded by the x-axis and a semi-undulation of 
the sine curve y = sin 2 x. (&) Find the area bounded by the x-axis and a 
complete undulation of the same curve, (c) Explain the result zero which 
the calculus gives for (6). (d) What is the number of square units bounded 
as in (&) ? 

14. Construct the figure, and show that, according to the calculus method 
of computing areas, the area between the curve whose equation is 12 y= (x — 1) 
( x _ 3) (x — 5), the x-axis, and the ordinates for which x = — 2 and x = 7, is 
— f | square units ; but that the 
actual number of square units in 
the figure thus bounded is 12£|. 

15. Find the area between the 
line 2 y — 5 x — 7 = 0, the x-axis, 
and the ordinates for which x = 2 
and x = 5, the axes being inclined 
at an angle 60°. 

fx=5 

Area AT OB -\ y sin 60° • dx 

= sin 60° f 5 (5x+ 1)dx 
— 63.65 square units. 

Note 4. In the light of the 
preceding examples attention may 
be again directed to the N.B. 

above. These examples also show : (1) the element of area may be 
chosen in various ways (compare Exs. 1, 4, 7, 9, 11) ; (2) the end values 
used in a problem may be chosen in different ways (see Ex. 3, Note 1) ; 
(3) the calculus method of computing areas should not be employed in a rule 
of thumb way, but with understanding and discretion (see Exs. 12, 13, 14). 




181.] AREAS OF CURVES. 319 

Note 5. Precautions to be taken in finding areas and computing 
integrals. Suppose that the area bounded by the curve y=f(x), the x- 
axis, and the ordinates at A and B for which x = a and x = b respectively, 
is required. If the curve has an infinite ordinate between A and B, or if 
the ordinate is infinite at A or B, or at both A and B, or if either or both 
the end values a and b are infinite, the area may be finite or it may be infinite. 
It all depends on the curve ; in one curve the area may be finite, in another 
curve it may be infinite. When infinite ordinates occur, either within or 
bounding the area whose measure is required, and also when the end-values 
are infinite, special care is necessary in applying the calculus to compute the 
area. The calculus method for finding areas and evaluating definite integrals 
can be used immediately with full confidence, only when the end values a 
and b are finite and when there is no infinite ordinate for any value of x from 
a to & inclusive. For illustrations showing the necessity for caution and 
special investigation in other cases see Murray's Integral Calculus, Art. 28, 
Exs. 3, 4, 5, 6, Art. 29 ; Gibson, Calculus, § 126 ; Snyder and Hutchinson, 
Calculus, Arts. 152, 155. 

Note 6. For the determination of the areas of curves whose equations 
are given in polar coordinates, see Art. 208. The beginner is able to proceed 
to Art. 208 now. 

EXAMPLES. 

16. Calculate the actual increases in area described in the Note and in 
Exs. 2, 4, Art. 67. 

17. Find the areas of the figures which have the following boundaries : 
(1) The curve y = x s and the line ky = x. (2) The parabola y 2 + 8x and 
the line x + y = 0. (3) The semi-cubical parabola y 2 = x s and the line 
y = 2 x. (4) The curves y 2 = x 3 and x 2 = 4 y. (5) The axes and the parab- 
ola Vx + Vy = Va. (6) The curve x 2 + 6y = and the line y + 3 = 0. 
(7) The curve (y + 4) 2 + (x + 3) 2 = and the line x + 6 = 0. (8) The 
hyperbola xy = 1 and the ordinates : (a) at x = 1, x = 7 ; (&) at x — 1, 
x = 15 ; (c) at x = 1 and x — n. (d) The hyperbola xy = k 2 and the ordi- 
nates at x = a and x = b. (And the z-axis in each case.) 

18. Find the area of the loop of the curve 8 y 2 = x 4 (3 + x). 

19. Show that the area of the figure bounded by an arc of a parabola and 
its chord is two-thirds the area of a parallelogram, two of whose opposite 
sides are the chord and a segment of a tangent to the parabola. 

[Suggestion : First take a parallelogram whose other sides are parallel to 
the axis of the parabola.] 

Ex. 20. Prove that the area of a closed curve is represented by 

^^f t -y^yt[ov^(xdy-ydx^ 

taken round the curve. (See Williamson, Integral Calculus, Art. 139 ; 
Gibson, Calculus, § 128.) 



320 



INTEGRAL CALCULUS. 



[Ch. XX. 



182. Volumes of solids of revolution. 

of the curve 



Suppose that the arc PQ 




Fig. 112. 



revolves about the cc-axis. It is required to find the volume 
enclosed by the surface generated by PQ in its revolution and 
the circular ends generated by the 
ordinates AP and BQ. (This is put 
briefly : the volume generated by PQ.) 
Let OA = a and OB = b. 

Suppose that AB is divided into 
any number of parts, say n, each equal 
to Ax. On any one of these parts, say 
LR, construct an ""inner" and an 
"outer" rectangle, as shown in Fig. 
112. Let G be the point {x, y), and K 
be the point (x + Ax, y + Ay). When 
PQ revolves about the a^axis, the inner rectangle GR describes a 
cylinder of radius GL {i.e. y), and thickness Ax. At the same 
time the outer rectangle KL describes a cylinder of radius KR 
{i.e. y + Ay), and thickness Ax. It is evident that the volume 
PQST is greater than the sum of the cylinders described by the 
inner rectangles, and is less than the sum of the cylinders described 
by the outer rectangles. That is, 

sum of outer cylinders > vol. PQST > sum of inner cylinders. 

The difference between the volume of the outer cylinders and 
the volume of the inner cylinders approaches zero when Ax 
approaches zero. Hence, 

vol. PQST— lim Aa ^ jsum of inner (or outer) cylinders J. 

That is, 

vol. PQST= lim Aa;i0 Jsum of cylinders like that generated 
by GR when x increases from a to b \ 

x=b 

= lim Ax = / {ttLG 2 - Ax) = tt I y^dx. 

(See Art. 166.) 



182.] 



VOLUMES OF E EVOLUTION. 



321 




The infinitesimal differential -n-y 2 dx, 
which is the volume of an infinitesimal 
cylinder of radius y and infinitesimal thick- 
ness dx, is called an element of volume. 

When PQ revolves about the ?/-axis the 
element of volume is evidently irx 2 dy. If 
the ordinates of P and Q are c and d respec- 
tively, the volume generated, 

d 



vol. PQTV=ir 



rv- 
Jy= 



ac^dy. 



Note 1. It is almost self-evident that the volume of the inner cylinders 
and the volume of the outer cylinders (Fig. 112), approach equality when 
their thickness Ax approaches zero. 

Note 2. See Art. 67(e). 

EXAMPLES. 

1. Find the volume generated by the revolution, about the ic-axis, of the 
part of the line 3 x + 10 y = 30 intercepted between 
the axes. 

The given line is AB. The element of volume 
is iry 2 dx. At B, x = ; at i, x = 10. Accord- 
ingly, the end-values of x are and 10. Hence, 



vol. cone ABC 



•£ 



y 2 dx 



I 



w/30 



=o " jo v 

94.248 cubic inches. 




Fig. 114. 



2. Verify the result in Ex. 1 by finding the volume of the cone in the 
ordinary way. 

3. Derive by the calculus the ordinary formula for finding the volume of 
a right circular cone having height h and base of radius a. (See Ex. 8.) 

4. (a) Find the volume generated by the revolution of the ellipse 

9 x 2 + 16 y 2 = 144 about the z-axis. (b) 
Find the volume bounded by a zone of the 
surface and the planes for which x = 2 and 
x = 3. 

The element of volume is wy 2 dx. 
(a) Vol. ellipsoid 




2w 



= 2 vol. ABB 

= ^f 4 (144 
16 Jo ^ 

= 150.8 cubic units. 



£ 



y 2 dx 

=o 

9x 2 )dx=48ir 



322 INTEGRAL CALCULUS. [Ch. XX. 

Or, vol. ellipsoid = v I y 2 dx = 150.8 cubic units. 

Jx=— 4 

(6) Vol. segment PQQ'P' = tt C =S y*dx = %ir = 17.08 cubic units. 

5. Find the volume generated by revolving the arc of the curve y = x 9 
between the points (0, 0) and (2, 8), about the y-a,xis. 

The arc is OA. The element of volume, taking any 
point P(x, y) on OA, is ttx 2 dy. Hence, 



vol. OAB ~ 7T f y 8 x 2 dy = tt f V 3 dy = ^-ir 

Jy=Q JO 



= 60.32 cubic units. 




The integral may also be expressed in terms of x. 
Thus, rx=2 

vol. OAB = tt\ x 2 dy. 

Jx=0 

Since y = x s , dy = 3 x 2 dx. 

.\ vol. OAB = Zt f V dx = - 9 / 7T = 60.32, as above. 



6. Find the volume generated by revolving about the ?/-axis the arc of 
the catenary x x 

between the lines a: = a and a; = — a. A CA' is the catenary ; A and A' are 
the points whose abscissas are a and — a respec- 
tively. The volume generated by revolving AC A' 
about OY is evidently the same as the volume gener- 
ated by revolving CA. The element of volume is 
ttx* dy. rx==a 

.-. vol. A CA' G = 7T J x 2 dy. (1 ) 

In this case it is easier to express the differential 
and the end-values in terms of x than in terms of 
y. From the equation of the curve it follows that 

x _x 

dy = \{ea — e «) dx. 
Hence (1) becomes vol. ACA'G=- f ° O 2 e« - x 2 e") dte. (2) 




Integration (by parts) of the terms in (2) gives 

vol. ACA'Q = l£(e + £-4 

2 V e 



.878 a\ 



182.] 



EXAMPLES. 



32a 



7. Find, by the calculus, the volume of the ring generated by revolv- 
ing a circle of radius 5 inches about a line distant 7 inches from the centre of 
the circle. 

Let C be the circle and ST the line. Choose 
for the x-axis the line passing through the centre 
at right angles to ST, and take OY for the 
y-axis. Then 

the equation of the circle is x 2 + y 2 = 25, 

and the equation of the line is x = 7. 

Through any point P(x, y) on the circle, draw 
Fig. 118. P'PM parallel to the x-axis. Suppose that PG, 

at right angles to PP', is of infinitesimal length 
dy. Then the rectangle P'G, on revolving about ST, generates an infini- 
tesimal part of the volume of the ring. The limit of the sum of these parts 
as y changes from B' to B, is the volume required. 

The volume generated by P'G = ir (WE 2 - PM 2 ) dy. 

Now P3I=7 -PB 

and 





Y 










R 






T 














R 


Vi'i 


,») 


M 








4 


°\ 













X 






/i 








if 






s 



V25 



P'31=7 + BP' = 7 +V25 



y 



. vol. generated by P'G = 28 w V25 - y 2 . dy 



[Or, 
as in Ex. 4 (a).] 



vol. of ring 
vol. of ring 



Jy=o 
= (^ 28tt-y/25 

Jy=-o 



28 TrV25-y 2 dy=350 tt 2 cubic units. 
(%=350 7r 2 cubic units, 



8. Find the volume of a cone in which the base is any plane figure of 
area A, and the perpendicular from the vertex to the base is h. 

9. Find the volume generated by revolving the arc BEC (Fig. 110) 
about the x-axis. 

10. Find the volume generated by the revolution of MTKN (Fig. 109) 
about the x-axis. 

11. Find the volume generated by the revolution of OBLM (Fig. 108) 
about the ?/-axis. / 

12. Find the volume generated by the revolution of ABLB (Fig. 107): 
(«) about the y-axis ; (&) about the x-axis. 

13. Find the volume generated by revolving the loop in Ex. 18, Art. 181, 
about the x-axis. 



824 INTEGRAL CALCULUS. [Ch. XX. 

14. Find, by the calculus, the volume generated by the revolution about 
the x-axis, of the part of each of the following lines that is intercepted between 
the axes, and verify the results by the ordinary rule for finding the volume 
of a cone : 

(l)3x + 4?/ = 2; (3) Ix + Sy + 20 = 0; 

(2) 2x-5y = 7 ; (4) 3x - 4y + 10 = 0. 

15. Find the volume generated by the revolution about the y-axis, of 
each of the intercepts in Ex. 14, and verify the result by the usual method 
of computation. 

16. Find the volume generated when each of the figures described in 
Ex. 17, (l)-(9), Art. 181, revolves about the x-axis. 

17. Find the volume generated when each of the figures in Ex. 16 
revolves about the ?/-axis. 

18. The figures bounded by a quadrant of an ellipse of semi-axes 9 
and 5 inches and the tangents at its extremities revolves about each tangent 
in turn : find the volumes of each of the solids thus generated. 

19. Find the volume of a sphere of radius a, considering the sphere 
as generated by the revolution of a circle about one of its diameters. 

Note 3. The volume of a sphere may also be obtained by considering the 
sphere as made up of concentric spherical shells of infinitesimal thickness. 
The volume of a shell whose inner radius is r and whose thickness is an infini- 
tesimal dr is (to within an infinitesimal of lower order) 4 irr 1 dr. Accordingly, 
volume of sphere = ( 4 irr 1 dr = f ira s . 

20. Find the volume generated by the revolution of the hypocycloid 
x 3+ y~3 = a"3 about the x-axis. (Ans. Yuz iraZ -) 

183. Derivation of the equations of curves. The equation of a 
curve or family of curves can be found when a geometrical prop- 
erty of a curve is known. Exercises of this kind constitute an 
important part of analytic geometry. For instance, the equation 
of a circle can be derived from the property that the points on 
the circle are at a given common distance from a fixed point. 
The' statement of a geometrical property possessed by a curve 
may involve derivatives or differentials. To derive the equation 
of the curve from this statement is, quite frequently, a difficult 
problem. There are a few simple cases, however, in which it is 
possible to find the equation of the curve by means of a knowl- 
edge of the preceding articles. A few very simple examples 
have been given in Art. 167. 



182, 183.] EQUATIONS OF CURVES. 325 

Note 1. It may be worth while merely to glance at more difficult prob- 
lems of this kind and at the text relating thereto, in Chapter XXVII. and in 
Murray's Introductory Course in Differential Equations, Chaps. V. and X. 
Also see Cajori, History of Mathematics, pp. 207-208, "Much greater than 
. . . integral of it." 

Note 2. It has been shown in Arts. 59, 62, that for the curve whose 
equation is /(x, y) = 0, rectangular coordinates, if (x, y) denotes any point 
on the curve and m is the slope of the tangent at (x, y) , then 

m = ( -H- ; subtangent = y — ; subnormal — y-^-. 
clx dy dx 

Note 3. It has been shown in Arts. 63, 64, that for the curve whose 
equation is f(r, 6) = 0, if (r, 6) denotes any point on the curve, \p the angle 
between the radius vector and the tangent at this point, and the angle 
which the tangent makes with the initial line, then 

tan^ = r— ; <f> = ^ + 0; 
dr 

polar subtangent = r 2 — ; polar subnormal = — • 
dr dd 

N.B. Draw the curves in the following examples. 

EXAMPLES. 

1. A curve has a constant subnormal 4 and passes through the point 
(3, 5) : what is its equation ? 

Here the subnormal, y-^- = 4. 

dx 

On using differentials, ydy = 4: dx. 



Integration gives 2- + Ci = 4 x + c 2 



whence *— = 4 x 4- k, in which Tc = c 2 — C\. 

A 

Since (3, 5) is on the curve, -^ = 12 + k, whence k-=\. 

v 2 1 

Accordingly, «- = 4sc + -, i.e. y 2 = 8 x + 1, is the equation. 

id A 

Note 4. In working these examples it is enlivening 
and helpful, to express the given conditions by means 
of a figure. This tentative figure can be corrected 
when fuller information is derived. Thus, for Ex. 1 
draw a curve passing through (3, 5), and at any point 
P(x, y) on this curve make the construction in 
Fig. 119. Fig. 119 showing the subnormal 4. Here Z MPN 

= ZHPT. Now tan MPN = -, i.e. - ? - = -• Then proceed as above. 

y dx y 




326 INTEGRAL CALCULUS. [Ch. XX. 

2. A curve has a constant subnormal and passes through the points 
(2, 4), (3, 8) : find its equation and the length of the constant subnormal. 

3. A curve has a constant subtangent 2, and passes through the point 
(4, 1) : find its equation. 

4. Determine the curve which has a constant subtangent and passes 
through the points (4, 1), (8, e) : find its equation and the length of the 
subtangent. 

5. Find the curve in which the length of the subtangent for any point 
is twice the length of the abscissa, and which passes through (3, 4). 

6. In what curves does the subnormal vary as the abscissa ? Deter- 
mine the curve in which the length of the subnormal for any point is pro- 
portional to the length of the abscissa, and which passes through the points 

(2, 4), (3, 8). 

7. In what curves does the slope vary as the abscissa ? Determine 
the curve in which the slope at any point is proportional to the length of the 
abscissa, and which passes through the points (0, 2), (3, 5). 

8. In what curves does the slope vary inversely as the ordinate ? 
Determine the curve in which the slope at any point is inversely proportional 
to the length of the ordinate and which passes through the points named in 
Ex.7. 

9. Determine the polar curves in which the tangent at any point 
makes with the initial line an angle equal to twice the vectorial angle. Which 

of these curves passes through the point (4, — ] ? 

10. Determine the polar curves in which the subtangent is twice the 
radius vector. Which of these curves passes through the point (2, C ) ? 

11. Determine the polar curves in which the subnormal varies as the sine 
of the vectorial angle, and which pass through the pole. 



CHAPTER XXI. 

INTEGRATION OF IRRATIONAL AND TRIGONOMETRIC 
FUNCTIONS. 

184. The integration of differential expressions involving irra- 
tional quantities and trigonometric quantities will now be con- 
sidered. Examples of this kind and methods of treating them 
have already been given in preceding articles. (See Art. 174, 
Art. 175, Exs. 10-18.) Only a few very special forms are dis- 
cussed in this book. 

Note. Chapter XIX. provides a good part of the knowledge of formal inte- 
gration sufficient for elementary work in physics and mechanics and for the 
ordinary problems in engineering. Accordingly, this chapter may be merely 
glanced at by those who have only a very short time to give to the study of 
the calculus and thus find it necessary to take on faith the results given in 
tables of integrals. 

INTEGRATION OF IRRATIONAL FUNCTIONS. 

185. The reciprocal substitution. This substitution, which some- 
times leads to an easily integrable form, has been shown in Art. 
177, Ex 6. Additional exercises are here appended. 

Ex. 1. Find f dx 

J x 2 Vz 2 - a 2 

Put x = — Then dx — dt ; and 

t P ' 

f — = - f tdt = — f (1 - a¥0"^(l - a 2 * 2 ) 

J* 2 Vx 2 -a 2 J Vl - aW 2 « 2j 



= I (1 _ «2,2Yi - ^E 



Exs. 2-9. Derive integrals 23, 26, 27, 39, 42, 43, 54 a, 59 a, 61 a, pages 
453-456. 

327 



328 INTEGRAL CALCULUS. [Ch. XXL 

Note. Trigonometric substitutions. Examples of a useful trigonometric 
substitution have been given in Art. 175, Exs. 4, 5. A differential expression 
in which Vet, 2 + x 2 occurs may sometimes be simplified for purposes of inte- 



gration by substituting atantf for x, and expressions containing Vx 2 — a 2 
by substituting a sec 6 for x. 

For instance, in Ex. 1 put x = a sec 6. Then dx = a sec 6 tan 6 dd ; and 

dx - l c cosede = Ume= Vx2 r< 



J «2v^ _ a* a 2 J 



186. Differential expressions involving Va + bx. By this is 

meant differentials in which the irrational terms or factors are 
fractional powers of a single form, a + bx. (In particular eases a 
may be and b may be 1 ; the irrational terms or factors are then 
fractional powers of x.) For preceding instances see Art. 175, 
Ex. 3, and Exs. 4, 10 at the end of Chapter XIX. 

If n is the least common denominator of the fractional indices 
of a + bx, the expression reduces to the form 



F(x, Va + bx) dx. (1) 

This can be rationalised by putting 
a + bx = z n . 



For then x — - and dx = - z n ~ x dz ; and, accordingly, ex- 

b b 



n F z -^,z)z^dz. 



pression (1) becomes 

b V b 

This is rational in z, and accordingly may be integrated by the 
preceding articles. 



Ex. 1. CJ^L^L. Ex. 4. f (3 + x) V(2 + x) 3 dx. 



Ex. 



1 + x^ 

f ^^. Ex. 5. f — 

J ^ + l J V-: 



+ 1 •> V2-z(7 + 5\/2-x) 



Ex. 3. f *** . Ex. 6. f V*-H + l ^ 

J \/(3 x - 2)* J Vx + 1 - 1 



186, 187.] IRRATIONAL FUNCTIONS. 329 



187 A. Expressions of the form F(x, a x' 2 + ax -f- b) dx. B. Ex- 
pressions of the form F(x, V — x 2 + ax + b) dx ; F(u, v) being a 
rational integral function of u and /. 

A. The first expression can be rationalised by putting 



X - 

dv- 


z 2 -b 
a + 2z 

_2(z 2 + az 


' + &) 


d* 




(a + 2 


zf 





Vic 2 + ckc + b = z — ®> (1) 

and changing the variable from x to & 

For, on squaring and solving Equation (1) for x, 

(2) 
From this, dx = m ^dz. (3) 



On substituting the value of x in (2) in the second member 
of (1), 

^+ax + b = z ' +az + b . 

a + 2 z 
Accordingly, 

F(x, Vx*+ax+b) dx becomes 2 F(±=h t+H^+HS *l + a * + b dz . 

\a-\-2z a+2z J (a+2zy 



This is rational in z, and, accordingly, may be integrated by 
preceding articles. 

Ex. 1. Find C 



Ind f 



Vx 2 — x + 1 
Assume Vx 2 — x + 1 = £ — x. 

2 2 -l 



From this, 



2^-1 



Then * = »(*— + !) *, 

(2 0-1) 2 

and Vx 2 - a: + 1 = - x = ^ ~ ^ + 1 . 

22-1 



330 INTEGRAL CALCULUS. [Ch. XXI. 

On substitution of these values in the given integral, 

C xdx =2 C 0-1 dz = hz + 8 + i 0g V2731 +c 

J Vx 2 -x + l J(?*-iy 4(2s-l) 

(See Art. 108.) 

_ x + Vx 2 - x + 1 3 



4 (2 x - 1 + 2 Vx 2 - x + 1) 



+ J log (2 x - 1 + 2 Vx 2 - x + 1) + c 

= J log (2 x - 1 + 2 Vx 2 - x + 1) + Vx 2 - x + 1 + k. 

(* = * + &) 

It happens that this is not the shortest way of working this particular 
example ; but the above serves to show the substitution described in this 
article. The integral may also be obtained in the following way ; this 
method is applicable to many integrals. 

r xdx r/i 2x-i + r 1 \ dx 

^Vx 2 -x + l * \' 2 Vx 2 -x + l 2 Vx 2 - X + 1 / 
= j"j (X 2 - X + l)~^(x 2 - x + 1) + i j" 



dx 



V(x - J) 2 + 



= Vx 2 - x + 1 + i log (x - I + Vx 2 - X + 1) + c 



= Vx 2 - x + 1 + I log (2 x - 1 + 2 Vx 2 - x + 1) + ci. 

Ex.2, f - £ -5)** =f( *~ 8 - 2 

J Vx 2 - 6 x + 25 J Wx 2 - 6 x + 25 V(x - 3) 2 + 16, 



= Vx 2 -6x + 25 - 2 log (x - 3 + Vx 2 - 6 x + 25). 

B. Suppose that — x 2 + ax + b = (x — p) (g — x). 

The second expression at the head of this article can be rational- 
ised by putting 



V— x 2 + ax + b, i.e. V(a? — p)(q — x) = (x — p) z, (3) 

and changing the variable from x to z. 

On squaring in (3), q — x = (a; — p) z 2 ; 

on solving for #, a? = ^— — ~ ; (4) 

1 + r 

2 # f p c/^ 

whence, on differentiation, dx = .. , — ~^- dz. 



187.] IRBATIONAL FUNCTIONS. 331 

Substitution of the value of x in (4) in the second member of 

Accordingly, 
F( x ■ y /-x i +ax+b)dx becomes 2 (©^»W^-±£ (ff~P)A *<& . 

This is rational in z, and, accordingly, may be integrated by 
preceding articles. 

Note 1. Instead of (8) the relation 



V(x-p)(q -x) = (q-x)z 
may be used. 



Note 2. If v ± px' 2 + qx + r occurs, it may be reduced to form A or 
B; thus, Vp J±x 2 + ^x+-- 



p p 

EXAMPLES. 



3. Find f- 



x 



Vl2 — x — x 2 



Put V12 - x - x 2 = V(x + 4) (3 - x) = (x + 4)2. 

From this, on squaring, 3 — x — (x + 4)s 2 . 
„ 3 - 4 z* 



On solving for x, 



1+z 



Accordingly, dx = ~ U \% Vl2 - x - x 2 = (x + 4) s = -^-- 

\1 + £-y 1 + z 

dx o C dz 1 , 2s — V3 



... r ** _ = 2 f 



1... 



xVl2-x-x 2 J 4 s 2 -3 2 V3 2z + VS 

_ _1_ 1qo . 2V3^-V3(x + 4), 
2V3 °2V3-x+V3(x+4) 

4. Solve Ex. 3, using the substitution Vl2 — x — x 2 = (3 — x) 0. 

5 f (2 x + 5) rfs 6 f (3 x - 4) die 

J V4 x 2 + 6 x +"il J V12 -4x-a; 2 

7 r <?x 

J x V12 - 4 x - x 2 
8 r (3 x - 4) c7x r 3 a; - 4 _ g 4 "I 

J x Vl2 - 4 x - x 2 L x x J 



332 INTEGRAL CALCULUS. [Ch. XXI. 

dx 

x Vx 2 + x + 1 



"■J 



r dx 10 f (x 2 + 2x-3)dx 

J v. -vA-2 a. /»•. -J. 1 ^ 



(Putx + 2 = 2.) 



(x + 2) Vx 2 + 4 x - 12 

2m+l 

Note 3. The integrands in integrals of the form ( xP(a + 6x 2 ) 2 dx in 

which m is any integer and p is an odd integer, positive or negative, can 
be rationalised by means of the substitution a + ox 2 = z 2 . Thus : 



■to f x 3 dx 




x 2 -a 2 = z\ 






J vx 2 — a 2 
Put 




Then 




xdx = z dz ; 






and (***-- 


(V 


+ a °-)dz = f (z 2 H 


^3a 2 ) 




= £-+!«! Vx2-«2. 



13. Find f — . (see Formula XXL, Art. 177): (1) Using the 

^ x Vx 2 — a 2 

substitution x = a sec 8 ; (2) using the substitution x = - ; (3) using the 

t 
substitution x 2 — a 2 = z 2 . (Show the equivalence of the various forms of 
the integral.) 

* 14. Show the truth of the statement in Note 3. 

188. To find J x m (a + bx n ) p dx. Here m, n, and p are constants, 

positive or negative, integral or fractional. The given integral, 
as will be shown in the working of examples, can be connected 
with simpler integrals in a particular way. By "a simpler inte- 
gral" is meant one that is simpler from the point of view of 
integration. For instance, if m = 5, the integral in which m = 3, 
other things being the same, is simpler ; if p = — f , the integral 
in which p = — \, other things being the same, is simpler. It will 
be found that the given integral can be connected with an integral in 
which the m is increased or decreased by n, or with an integral in 
which the p is increased or decreased by 1 ; i.e., with one or other 
of the four integrals : 



Cx m+n (a + bx n ) p dx, Cx m (a + bx n ) p+1 dx, 

Cx m ~ n (a + bx n ) p dx, Cx m (a + bx n ) p ~ l dx. 



(a) 



187, 188.] IRRA T10NAL FUNCTIONS. 333 

When one of these four integrals is chosen, a relation between 
it and the required integral can be expressed in the following 
way: 

Form a function of x in which the x outside the bracket has an 
index one greater than the least index of the corresponding x in the 
required and the chosen integrals, and in which the bracket has an 
index one greater than the least index of the bracket in those integrals. 
Give the function thus formed an arbitrary constant coefficient and 
give the chosen integral an arbitrary constant coefficient-; equate the 
sum of these quantities to the required integral. The value of the 
arbitrary coefficients can then be determined. 

For example, let 

j x m (a + bx n ) p dx be connected with J x m (a+bx n ) p - 1 dx. 

The function formed by the rule is x m+1 (a + bx n ) p . Put 
Cx m (a + bx n ) p dx = Ax m+ \a + bx n ) p + B Cx m (a + bx n ) p - x dx, (1) 

in which A and B are arbitrary constants. 

It is now necessary to find such values for A and B as will 
make (1) an identical equation. 

In order to determine A and B, take the derivatives of both 
members of (1), simplify, and then equate coefficients of like 
powers of x. Thus, on differentiating the members of (1), 

x m (a + bx n ) p = A(m + l)x m (a + bx n ) p + Ax m+1 p(a + bx n ) p -hibx n ~ x 

+ Bx m (a + bx n ) p ~\ 

On division by x m (a + bx 71 )^ 1 , and simplification, 

a + bx n = Ab{m + np + l)x n + Aa(m + 1) 4- B. 

On equating coefficients of like powers of x and solving for A 
and B, 

A = * , B= an P - . 

m + np + 1 m-r-np-\-l 



334 INTEGRAL CALCULUS. [Ch. XXI. 

The substitution of these values in (1) gives 



f 



x m (a + bx n )Pdx = 



m 4- np 
anp 



m + np + 



j f « m (a + bx n )P-* dx. 



On connecting the required integral with each of the other 
integrals in (a) and proceeding in a similar manner, the results 
(1), (2), (4), page 451, are obtained. The deduction of them is 
left as an exercise for the student. 

Note 1. Formulas 1-4, page 451, are examples of what are usually termed 
Formulas of Reduction. Frequently integrals are obtained by substituting 
the particular values of m, n, p in these formulas of reduction. To memorize 
such formulas is, however, a waste of energy ; it is better, at least for 
beginners, to integrate by the method whereby these formulas have been 
obtained. 

Note 2. It will be observed that some of these formulas fail for certain 
values of m, n, p ; viz., when m + np + 1 = 0, when m = — 1, and when 
p = — 1. Other formulas or other methods may be applied in each of these 
cases. 

Note 3. Its success may be regarded as one proof of the above method. 
In the large majority of text-books on calculus, formulas 1, 2, 3, 4, page 451, 
are derived in a straightforward way by integration by parts. For this 
derivation see almost any calculus, e.g. Murray, Integral Calculus, Appendix, 

Note B. For other formulas of reduction for \ x m (a + bx n y dx, obtained 

by the method of "connection" or "arbitrary coefficients," see Edwards, 
Integral Calculus, Art. 82, and integrals 5, 6, page 452. 

EXAMPLES. 

1. Find f dx i.e. f x~ 2 (x 2 - a^dx. (See Ex. 1, Art. 185.) 

J x 2 Vx 2 — a 2 ^ 

Here m = — 2, n = 2, p = — \. The best integral to connect with is 

C dx 
obviously the integral in which the m is raised by 2, viz. i . On 

J Vx 2 - a 2 

making the connection according to the directions given above, 

(1) f x- 2 (x 2 - a 2 )~% dx = Ax~\x 2 - a 2 y + B f(x 2 - a 2 )~? dx. 

It is now necessary to find such values for A and B as will make this 
equation an identical equation. 



188.] IRRATIONAL FUNCTIONS. 335 

On differentiation, and equating the derivatives, 

x -2 ( X 2 _ rt 2)"i _ _ Ax -2 ( X 2 _ 2) i + A (a-2 _ a 2)"4 + £ ^2 _ a 2)~2\ 

Oa simplifying, by multiplying through by x 2 (x 2 — a 2 ) 2 ", 

1 = - ^l(x 2 - a 2 ) + ^lx 2 + Bx\ 

On equating coefficients of like powers of x, 

B = and ^4a 2 = 1 ; whence A——. 

a 2 

On substitution of these values of A and B in (1), 

C dx _ Vx 2 - a 2 
Jx 2 Vx 2 - a 2_ « 2;c 

2. Find f ^ x , i.e. (*x 3 (x 2 - a 2 ) _2 ^x. (See Ex. 12, Art. 187.) 
J Vx 2 - cfi J 

Here m = 3, n = 2, p = — J. It will obviously be an advantage to lessen 

m. Accordingly, let connection be made with ( x (x 2 — a 2 )" 2 dx. On doing 
this in the way described, 

(1) f x 3 (x 2 - a 2 )"- dx = Ax 2 (x 2 - a 2 )? + B f x (x 2 - a 2 )"- dx. 

It is now necessary to find such values for A and B as will make this an 
identical equation. 

On taking the derivatives and equating them, 

x s ( X 2 _ a 2yl _ 2 Ax (x 2 - a 2 ) 2 ^ + ^x 3 (x 2 - a 2 )~^ + .Bx (x 2 - a 2 ) _2 \ 

On simplifying, by dividing through by x (x 2 — a 2 )" 2 , 

x 2 = 2 yl (x 2 - a 2 ) + ^lx 2 + B. 

On equating coefficients of like powers of x, and solving for A and J5, it is 
found that A = i, £ = § a 2 . 
Substitution in (1) gives 



J Vx 2 - a 2 J Vx 2 - a? 



+ 2 a 2 ) Vx 2 



3. Find f ^e , i.e. f (x 2 + a 2 )-*dx. 

J (x 2 + a 2 )*' J v y 

Here m = 0, n = 2, p = — k. In this case 
oceeding according to the rule, 

(1) f (x 2 + a 2 )~ k dx = Ax(x 2 + a 2 )-^ 1 + B f (x 2 + a 2 )-*+M*. 



Here m = 0, n = 2, j? = — k. In this case it is better to increase p. On 
proceeding according to the rule, 



336 INTEGRAL CALCULUS. [Ch. XXI. 

On differentiation, simplification of the resulting equation by division by 
(x 2 + a 2 )-*, equating coefficients of like powers, solving for A and B, and 
substitution of their values in (1), it will be found that 

(—** = l I- * + (2k-S)( *? 1. 

J (a 2 + a 2 ) k 2 a\k - 1) I (x 2 + a 2 )*- 1 v J J (x 2 + a 2 )*- 1 / 

4. Derive j Va 2 — x 2 dx by this method. (See Ex. 5, Art. 175, Ex. 5, 
Art. 176.) J 

5. Do Ex. 16, Art. 177, by this method. 

Note 4. It is sometimes necessary to repeat the operation of reduction 
two or more times. 

6. Derive integrals 21, 22, 23, 28, 30, 35, 40, 41, 42, 44, pages 453-454, 
and others of the collection. 



7. Derive integrals 48, 53, 54, 55, 57, pages 455-456 [ V2 ax ± x- 
x?(2a ± x)?]. (Compare Exs. 6, 7, and Exs. 2-9, Art. 185.) 



8. Derive formulas 1-6, page 451, 

x 4 3 a 2 x 3 



9. Find C^~^ dx= _^-**y 
J x 4 



f — dx = - { 3 a 4 sin-i £ - x(2 x 2 + 3 a 2 ) Va 2 - x 2 \ . 

10. Using integrals 1-4 as formulas of substitution for the values of m, w, 
p, a, 6, derive some of the integrals 21-30, 37-46, 53-61, pages 453-456. 

Note 5. On the integration of irrational expressions also see Snyder and 
Hutchinson, Calculus, Arts. 129-131, 139, 140. These articles convey valu- 
able additional information, and, in particular, Art. 139 gives an interesting 
geometrical interpretation concerning the rationalisation of the square root 
of a quadratic expression. Also see the references given in Art. 192, Note 2. 



INTEGRATION OF TRIGONOMETRIC FUNCTIONS. 

N.B. On account of the numerous relations between the trigonometric 
ratios, the indefinite integral of a trigonometric differential can take many 
forms. 

189. Algebraic transformations. A differential expression in- 
volving only trigonometric ratios can be transformed into an 
algebraic differential by substituting a variable, t say, for one of 
the trigonometric ratios. The algebraic differential thus obtained 
may possibly be integrated by some method shown in the preced- 
ing articles. Knowledge as to what substitution wili be the most 



189, 190.] TRIGONOMETRIC FUNCTIONS. 337 

convenient one to make in a given case can best be acquired by 
trial and experience. Illustrations of this article have already 
been met in Art. 175, Exs. 10, 11, 16, 17. 

Ex. 1. See exercises just referred to. 

Ex. 2. Do Exs. 1-5, 7-9, Art. 190, making algebraic transformations. 

190. Integrals reducible to J F(u) du, in which u is one of the 

trigonometric ratios. 

{a) I sin n x dx and J cos w x dx are thus reducible when n is an 
odd positive integer. For 

I sin n x dx = I sin" -1 x • sin xdx = — I (1 — cos 2 x) ~ d (cos x). 

The latter form can be expanded in a finite number of terms, n ~ 

being an integer, and then integrated term by term. | cos n x dx 
can be treated similarly. 

EXAMPLES. 

1. | cos 5 x dx = ( cos 4 x • cos x dx = \ (1 — sin 2 x) 2 d (sin x) 

= 1(1—2 sin 2 x + sin 4 x) d (sin x) = sin x — § sin 3 x + \ sin 5 x + c. 

2. ( sin 3 x dx, ( cos 3 x dx, \ sin 5 x dx. 

(&) I sin n x cos m a? da? is thus reducible when either n or m is a 
positive odd integer. 

3. \ sin 3 x cos 2 " x dx = ( sin 2 x cos 2 x sin x dx 

/* 5 /• 5. 9 

= — \ (1 — cos 2 x) cos 2 x d (cos x) = — I (cos 2 x — cos 2 x) d (cos x) 
1 u. 

= — } COS 2 X + j^ COS 2 X + c. 

4. (1) f-f^dx, (2) fcos 5 xsintxdx, (3) f™^, 

*^ Vcosx ^ ^ Vsinx 

f 4) l cos ¥ x sin 3 x dx. 

Note. Case (a) is a special case of (6). 



338 INTEGRAL CALCULUS. [Ch. XXI. 

(c) I sec n oc dx and I cosec M a? <?a? are ^us reducible when n is 
a positive even integer. 

5. 1 cosec 6 xdx — \ cosec 4 x • cosec 2 xdx = — 1(1 + cot 2 x) 2 d (cot a;) 

= - cot x (1 +| cot 2 X + £ cot 4 x). 

6. Show the truth of statement (c). 

7. (1) (sec* xdx, (2) ( cosec 4 xdx, (3) I sec 6 xdx. 

(<J) | tan m 05 sec™ a? da? and I cot™ a? cosec™ a? da? are £ftws reduci- 
ble when n is a positive even integer, or when m is a positive odd 
integer. 

8. Show the truth of statement (d). 

9. (1) \ tan 2 x sec 4 x dx, (2) j sec 6 x Vtan x dx, (3) \ tan ¥ x sec 4 x dx, 
(4) ( tan 5 x sec 3 x dx, (5) ( cot 3 x Vcosec x dx, (6) cot 5 x cosec 3 x dx. 

191. Integration aided by multiple angles. It is shown in 
trigonometry that 

sin u cos u = ^ sin 2 u, 

sin 2 w = \ (1 — cos 2 w), 

cos 2 tt = i(l + cos 2 w). 

Accordingly, if n and m are positive even integers, sin w x, cos" a?, 
and sin n xcos TO £ can be transformed into expressions which are 
rational trigonometric functions of 2 x. Differential expressions 
involving the latter are, in general, more easily integrable than 
the original differential expressions in x. 

Ex. 1. fcos 4 xdx = ({^(1 + cos2x)} 2 dx = \\ (1 + 2 cos 2 x + cos 2 2x) dx. 

Now I 2 cos 2 x dx = sin 2 x, and I cos 2 2 xdx = £ i (1 + cos4x) dx = 

|(x + £ sin 4 x) . .-. ( cos 4 x dx = f x -f J sin 2 x -f ^ sin 4 x -f c. 

Ex. 2. t sin 2 a: cos 2 x dx = J j sin 2 2 x dx = i J (1 — cos 4 x) dx 

= i x ~ "h sm 4 x + c. 
Ex.3. (1) fsin 4 xdx, (2) fcos 6 xdx, (3) f sin 4 x cos 2 xdx, 

(4) | sin 3 x cos 3 x dx, (5) ( sin 4 x cos 4 x dx. 



190, 192.] TRIGONOMETRIC FUNCTIONS. 339 

192. Reduction formulas. There are several formulas which 
are useful in integrating trigonometric differentials. A few of 
them are deduced here ; the deduction of the others is left as an 
exercise for the student. 

(a) To find A: I sin n xdx, and B: I cos n xdx, when n is any 
integer. 

A. Integrate by parts, putting 

dv = sin x dx ; then u = sin" -1 x, 

v = — cos x, du = (n — 1) sin n ~ 2 # cos x dx. 

.-. I sin n # dx = — sin n_l £ cos x -\- (n — 1) I sm n ~ 2 x cos 2 xdx 

= — sin 71-1 x cos x + (?i — 1) I sin n-2 # (1 — sin 2 #) dx 

= — sin n_1 x cos x -+- (n — 1) I s\n n ~ 2 xdx — (n — 1) I sin n xdx l 

From this, on transposition and division by n, 

/• n 7 sin w_1 a; cos a; . n — 1 /* . n « 7 ^ x 

sm rt a? cto = 1 | sin" -2 x dx. (1) 

n n J 

This is a useful formula of reduction when n is a positive 
integer. From it can be deduced a formula which is useful when 
the index is a negative integer. For, on transposition and division 

71 — 1 

by j formula (1) becomes 

/■ n -o 7 sin n_1 x cos x , n /~ • „ 7 
sm n - a? cic = 1 I sm n x dx. 
n — 1 n — U 

This result is true for all values of n, and, accordingly, for 
n =z JN+ 2. On putting JV + 2 for w, this becomes 

/• - , sin^ +1 x cos x , N + 2 T . „, , , /ON 

sin^sc die = -L- - I sm^ +2 ic dx. (2) 

If iVis a negative integer, say — m, (2) may be written 

/ dx _ _ 1 cos£ m — 2 r_dx__ ,„\ 

sin m ic m — 1 sin m-1 # m — 1 J sin m_2 £c 



340 INTEGRAL CALCULUS. [Ch. XXI. 

In the above way calculate the following integrals : 

Ex. 1. (1) (s\n 2 xdx, (2) \sin 3 xdx, (3) \sin±xdx, (4) (sm 5 xdx. 

Ex. 2. (1) f-^-, (2) f r ^-, (3) C-^L.. 
Jsin 2 sc v J sin* a; v J J sin* a; 

Ex. 3. Compare the results in Exs. 1, 2, with those obtained for these 
integrals by methods of the preceding articles. 

B. Similarly to A there can be deduced results 69, 71, page 457, 
for B. Formula 69 is useful for positive indices, and 71 for 
negative indices. 

Ex. 4. Deduce formulas 69 and 71. 

Ex. 5. (1) (cos*xdx, (2) fcos 5 z<to, (3) f-^-, (4) f-^_. 
J J Jcos 4 x J cos 5 a; 

Compare results with those obtained by methods of preceding articles. 
(b) To find J sec n x dx when n is a positive integer greater than 1. 

Put sec 2 xdx = dv; then sec n ~ 2 x = u, 

tan x = v, (n — 2) sec n_2 x tan xdx= du. 
.*. I sec' 1 x dx = sec n_2 x tan x — (n — 2) I sec n_2 x tan 2 x dx. 

From this, on substituting sec 2 a; — 1 for tan 2 a;, and solving 
for isec n xdx, 

J 7 sec n ~ 2 x tan x , n — 2 C n-2 ^ 
sec n # dx = ! I sec n 2 x dx. 
n — X n — \J 

Similarly, result 73 for I cosec" x dx can be obtained. 

Ex. 6. (1) fsec 3 xdx, (2) \setfxdx, (3) setfxdx. 

Ex. 7. (1) (csc^xdx, (2) (cstfxdx, (3) csc b xdx. 
Ex. 8. Derive formula 73. 

Ex. 9. Erom formulas 72 and 73 derive formulas for \sec n xdx and 
i cosec" x dx which are applicable when n is a negative integer. 
[Suggestion : Use method employed in deducing formulas 70 and 71.] 



192.] 3 RIGONOMETRIC FUNCTIONS. 341 

(c) To find I ta)i n xdx, in which n is a positive integer greater 
than 1. 

J tan n x dx = j tan" -2 x tan 2 xdx= I tan n-2 a; (sec 2 a; — 1) cfa? 

= I tan" -2 x d (tan #) — | tan" -2 a; c?a? 

= tan^£_ f tQt11 *-*xdx. 
n-1 J 

Similarly can be shown result 75 for j cot n xdx. 

When n is negative, say — m, then lta.Ti n xdx = \aot m xdx, 
and ltan n .i*d.y can be expressed in cotangents by formula 75. 

Formulas applicable to cases in which n is negative, can be 
deduced from formulas 74 and 75, by the method used in 
deducing formulas 70 and 71. 

Ex. 10. Deduce Formula 75, and formulas applicable to t tan" x dx and 
I cot" x dx when n is negative. 

Ex. 11. (1) ftan 3 a:^x, (2) cotfxdx, (3) ftan 4 £<£c, (4) (t&ifixdx. 

(d) I sin™ l x cos n oc dx. When m and n are integers, reduction 

formulas can be derived for this integral in a manner similar to 
that used in Art. 188 ; that is, by 

(i) Connecting it with each of the four integrals in turn, viz. : 
I sin™ -2 x cos n x dx, ) sin" 1 x cos n-2 x dx, 

I sin m+2 x cos w x dx, l sin M x cos n+2 x dx ; 

(ii) Forming a new function by giving sin x and cos x each an 
index one greater than the lesser of its indices in the required 
integral and the integral with which it is connected, and taking 
the product ; 

(iii) Giving the connected integral and this newly formed 
function each an arbitrary coefficient, and equating their sum to 
the required integral ; 



342 INTEGRAL CALCULUS. [Ch. XXI 

(iv) Determining the value of these coefficients by proceeding 
as in Art. 188. 

The derivation of these reduction formulas is left as an exercise 
for the student j they are given in the set of integrals, Nos. 76-79.* 

Ex. 12. Deduce formulas Nos. 76-79 by the methods outlined above. 
Ex. 13. Deduce the formulas in Ex. 12 by integrating by parts. 
Ex. 14. Apply these formulas to finding the following integrals : 

(1) Tsui 2 * cos 2 z <£c; (2) fcos 4 x sin 2 x ; (3) C^^dx. 

Ex. 15. Deduce the integrals in Ex. 14 by the method outlined in (d). 
Note 1. When m + n is a negative even integer, the above integral can 
be expressed in the form f /(tan x)d(t&n x). 

Ex. 16. C^l dx = C*™-* . _JL_ . dx = f tan 3 x sec* x dx 

J cos 7 X J COS 3 X COS 4 X J 



J 



COS 3 X COS 4 X 

tan 3 x (1 + tan 2 x) d tan x = ^(6 + 4 tan 2 x) tan 4 x. 

Ex.17. (1) (^dx, (2) f^efc, (3) f™f*<fc. 

J sin 8 x J cos x J sin 6 x 

Note 2. Special forms. Integrals 80-87 are occasionally required. Eor 
their deduction see Murray, Integral Calculus, Arts. 54-57, or other texts. 
It will be a good exercise for the student to try to deduce these integrals him- 
self. Eor a fuller discussion of the integration of irrational and trigonometric 
functions see the article Infinitesimal Calculus (Ency. Brit., 9th edition), 
§§ 124 on ; also see Echols, Calculus, Chap. XVIII. 

Note 3. On integration by infinite series. See Art. 197. 

Note 4. Elliptic integrals. Elliptic functions. The algebraic inte- 
grands considered in this book give rise only to the ordinary algebraic, 
circular, and hyperbolic t functions. (The two last named are singly periodic 
functions.) Certain irrational integrands give rise to a class of functions 
treated in higher mathematics, viz. the elliptic (or doubly periodic) functions. 
The term elliptic functions is somewhat of a misnomer; for the elliptic 
functions are not connected with an ellipse in the same way as the circular 
functions are connected with the circle, and the hyperbolic functions with 
the hyperbola. The elliptic integrals derived their name from the fact that 
an integral of this kind appeared in the determination of the length of an 
arc of the ellipse. Out of the study of the elliptic integrals arose the modern 

* These formulas are derived in Murray, Integral Calculus, Art. 51, and 
Appendix, Note C. Also see Edwards, Integral Calculus, Art. 83. 
t See Appendix, Note A. 



192.] TRIGONOMETRIC FUNCTIONS. 343 

extensive and important subject of elliptic functions ; this accounts for the 
term elliptic in the name of these functions. The student may take a glance 
forward and extend his mathematical outlook by inspecting Art. 174, Note 4; 
Cajori, History of Mathematics, pages 279, 347-354 ; the section on elliptic 
integrals in the article mentioned in Note 2, in particular, §§ 191, 192, 204, 
205, 206, 219, 220 ; W. B. Smith, Infinitesimal Analysis, Vol. I., Arts. 123-125 ; 
Glaisher, Elliptic Functions, pages 6, 175, etc. 



EXAMPLES. 

1. Derive integrals Nos. 80-82, 85-87. 

2. Derive several of the integrals 18-30, 36-46, 53-65. 
Vt -,. /on C vdv , \ C dx 



3. (1) C^Ldt. (2) f— **! (3) r 

(4) f xd x (5) V4x - X * dx. (6) f (2_* +!)<*« 



(1 + a 2 ) Vl - 4 x 2 ^ ^ Vx 2 + 3 x + 5 



dx 



(7) r c ^+ 1 )^ . (8 ) r * (9) f_* 

*Wx 2 + 3x + 5 ^(^ + l)V^TI ■) («*-. 16 ) 

dx ,,„ N fVOx 



( io) r_*5 — (id r — ^ (12) f™ 



C?X. 



4. Derive the following integrals 



(1) I -J a + x dx = a sin" 1 - - Va 2 — x 2 . 
J *a — x a 

(2) CJ^-±J1 dx = - V(a + x)(5 - x) - (a + &)sin-iJ^^. 
J > 6 — x 'a + 5 

(3) fJ|^ dx = V(a-x)(6 + x) + (a + 6) sin-i \/^| • 

(4) f \i J- 1 ^ ^ = V(a + x) (b + x) + (a - b) log ( Va~+^ + VH^) . 
J *b -\- x 

(5) f * = 2sin-ijLEJ. 
J V(x-a)(6-x) >6-a 

5. Show that, if f(u, v) is a rational function of u and v, and m and n are 

TO 

integers, then /{x 2 , (a + 5x 2 )' l }xdx can be rationalised by means of the sub- 
stitution a + 6x 2 = z n . (Ex. 14, or Note 3, Art. 187, is a particular case of this 
theorem.) n 

6. Show that (1) f 2 sin 2 "* dx = 1 • 3 • 5 - (2m - 1) _ tt 

> 2 -4-6 ...2m 2 ' 



< 2 > X s 



sm 2m + l xdx = — - — — — — — (m being an integer). 

3 • 5 • 7 ••• (2 m + 1) 



CHAPTER XXII. 



APPROXIMATE INTEGRATION. MECHANICAL. 
INTEGRATION. 

193. Approximate integration of definite integrals. It has been 
shown in Arts. 165, 166, 168, that: (a) the definite integral 

I f(x)dx may be evaluated by finding the anti-differential of 

f(x)dx, <£(#) say, and calculating <f>(b) — <£(«) ; (b) this last num- 
ber is also the measure of the area of the figure bounded by the 
curve y=f(x), the #-axis, and the two ordinates for which x = a 
and x = b. In only a few cases, however, can the anti-differential 
oif(x)dx be found; in other cases an approximate value of the 
definite integral can be obtained by making use of fact (b). Thus, 
on the one hand the evaluation of a definite integral serves to 
give the measurement of an area ; on the other hand the accurate 
measurement of a certain area will give the exact value of a defi- 
nite integral, and an approximate determination of this area will 
give an approximate value of the integral. The area described 
above may be found approximately by one of several methods; 
two of these methods are explained in Arts. 194 and 195. 

194. Trapezoidal rule for measuring areas (and evaluating definite 
integrals). Let the value of the definite integral J f(x)dx be 

required. Plot the curve 
y =f(x) from x = a to x = 5. 
Let OA = a, OB = b, and draw 
the ordinates AP and BQ. By 
Art. 166, the measure of the 
area APQB is the value of the 
required integral. An approxi- 
mate value of the area APQB 
u i y can be found in the following 



P rt -iQ 




d x 



344 



193,194.] APPROXIMATE INTEGRATION. 345 

way. Divide the base AB into n intervals each equal to Ax, and 
at the points of division A h A 2 , A 3 , •••, erect ordinates A X P X , 
A 2 P 2 , A 3 P 3 , ••-. Draw the chords PI\, P X P 2 , P 2 P 3 , ••-, thus 
forming the trapezoids AP 1} A X P 2 , A 2 P 3 , •••• The sum of the 
areas of these trapezoids will give an approximate value of 
the area of APQB. 

Area AP X == \ (AP + A^) Ax, 

area A X P 2 = i (A^ + A 2 P 2 ) Ax, 

area A 2 P 3 = ± (A 2 P 2 + A 3 P 3 ) Ax, 



area A n _ x Q = i (A-A-i + BQ) Ax. 
.-. area of trapezoids = (i AP -f ^Px + -4 2 P 2 + ••• + A n _ x P n _ x 
+ iBQ)Ax. 
This result may be indicated thus : 

area trapezoids = (| + 1 + 1 + ... + 1 + 1) Ax, 

in which the numbers in the brackets are to be taken with the 
successive ordinates beginning with AP and ending with BQ. 

Note. It is evident that the greater the number of intervals into which 
b — a is divided, the more nearly will the total area of the trapezoids come 
to the actual area between the curve and the x-axis, and, accordingly, the 
more nearly to the value of the integral. See Exs. 1, 2. 

EXAMPLES. 

/»12 

1. Find \ x 2 dx, dividing 12 — 1 into 11 equal intervals. 
Here each interval, *Ax, is 1. Hence, approximate value 

= (i • I 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 + 8 2 + 9 2 + 10 2 + ll 2 + \ • 12 2 ) = 5771 

The value of I x-dx = — + c = 575#. The error in the result ob- 
h L3 Ji 3 

tained by the trapezoidal method is thus, in this instance, less than one- 
third of one per cent. 

2. Show that if 22 equal intervals be taken in the above integral, the 

approximate value found is 576.125. 

rv\ 

3. Show that on using the trapezoidal rule for evaluating \ x 2 dx, 

if 10 intervals be taken, the result is If units more than the true value, 
and if 20 intervals be taken, the result is -^ of a unit more than the true 
value. 



346 



INTEGRAL CALCULUS. 



[Ch. XXII. 



4. Explain why the approximate values found for the integrals in 
Exs. 1, 2, 3, are greater than the true values. 

/"320 

5. Evaluate I cos x dx by the trapezoidal rule, taking 10' intervals. 

(Ans. .0148. The calculus method gives .0149.) 

/•320 

6. Evaluate t sin x dx, taking 30' intervals. 

(Ans. .0506. Calculus gives .0508.) 

/*35o 

7. Evaluate \ cos x dx, taking 1° intervals. 

(Ans. .1509. Calculus gives. 1510.) 

195. Parabolic rule* for measuring areas and evaluating definite 
integrals. Let the area and the integral be as specified in Art. 
194. For the application of the parabolic rule, the interval AB 
is divided into an even 
number of equal intervals 
each equal to Asc, say. The 
ordinates are drawn at the 
points of division. Through 
each successive set of three 
points (P, P„ P 2 ), (P 2 , P 8 , 
P 4 ), ••-, are drawn arcs of 
parabolas whose axes are 
parallel to the ordinates. The area between these parabolic arcs 
and the #-axis will be approximately equal to the area between 
the given curve and the o>axis. The area bounded by one of these 
parabolic arcs and the #-axis, and a pair of ordinates, say the 
area of the parabolic strip APP 1 P 2 A 2 , will now be found. 

Parabolic strip APP X P 2 A 2 = trapezoid APP 2 A 2 + parabolic 

segment PPiP 2 . (1) 

Now the parabolic segment PP X P 2 

= two-thirds of its circumscribing 
parallelogram PP'P' 2 P 2 .-f (2) 




Fig. 121. 



* This rule, which is much used by engineers for measuring areas, is also 
known as Simpson's one-third rule, from its inventor, Thomas Simpson 
(1710-1761), Professor of Mathematics at Woolwich. 

t See Art. 181, Ex. 19. 



195.] PARABOLIC RULE. 347 

Area trapezoid APP 2 A, = J AA 2 (AP +- A 2 P 2 ) ; 

area PP'P' 2 P 2 = area AP'P' 2 A 2 - area ^LPP 2 .4 2 

= 2.iii 2 .i 1 A-iM(iP 

+ AP 2 > (3) 

Hence, by (1), (2), and (3), area parabolic strip APP 1 P 2 A 2 

= (AP+±A 1 P 1 + A 2 P^. 
Similarly, area of next parabolic strip A 2 P 2 P S P^A A 

= (A 2 P 2 + ±A s P s + A 4 P i )^; 

and so on. Addition of the successive areas gives total area of 
parabolic strip =(AP + ± A X P, + 2 A 2 P 2 + 4 A 3 P 3 

+ 2A i P i +-+BQ)^- 
This result may be indicated thns : 

Total parabolic area = (1+4+2 + 4 + . .. + 2 + 4 + 1) ~, (4) 

o 

in which the numbers in the brackets are understood to be taken 
with the successive ordinates beginning with AP and ending 
with BQ. 

EXAMPLES. 

rio 
1. Find \ x 3 dx, taking 10 equal intervals. 

Here, each interval = 1. Hence, the result by (4) 

= (1 • 03 + 4 • 13 + 2 • 23 + 4 • 3 3 + 2 • 43 + 4 • 53 + 2 • 6 3 + 4 • 7 3 

+ 2 • 8 3 + 4 • 93 + 1 • 10 3 ) x J = 2500. 

-a* -no 



t r 4 -| 10 

— + c = 2 



2500. 



2. Calculate the above integral, using the trapezoidal rule and taking 
10 equal intervals. 

f n 

3. Evaluate \ x % dx, both by the trapezoidal and the parabolic rules, 

taking 10 equal intervals. 

4. Evaluate Ex. 1, Art. 194, by the parabolic rule. Why is the result 
the true value of the integral ? 

5. Show that there is onlv an error of 14 in 20,000 made in evaluating 
/no 

I x i dx by the parabolic method, when 10 intervals are taken. 



348 INTEGRAL CALCULUS. [Ch. XXII. 

6. Find the error in the evaluation of the integral in Ex. 5 by the trape- 
zoidal method, when 10 intervals are taken. 

7. Evaluate the integrals in Exs. 6, 7, Art. 194, by the parabolic rule. 
Note. For a comparison between the trapezoidal and parabolic rules, for 

a statement of Dnrand's rule, which is an empirical deduction from these 
two rules, for a statement of other rules for approximate integration, and 
for a note on the outside limits of error in the case of the trapezoidal and 
parabolic rules, see Murray, Integral Calculus, Arts. 86, 87, Appendix, Note 
E, and foot-note, page 186. 

196. Mechanical devices for integration. The value of a definite 
integral may be determined by various instruments. Accordingly, 
they may be called mechanical integrators. Of these there are 
three classes, viz. planimeters, integrators, and integraphs. These 
instruments are a great aid to civil, mechanical, and marine 
engineers. The area of any plane figure can be easily and accu- 
rately calculated by each of these mechanisms. Their right to be 
termed mechanical integrators depends on the facts emphasised 
in Arts. 166, 168, 193-195 ; the facts, namely, that a definite inte- 
gral can be represented by a plane area such that the number of 
square units in the area is the same as the number of units in the 
integral, and hence that one way of calculating a definite integral 
is to make a proper areal representation of the integral and then 
measure this area. 

Planimeters, which are of two kinds, viz. polar planimeters and 
rolling planimeters, are designed for finding the area of any plane 
surface represented by a figure drawn to any scale. The first 
planimeter was devised in 1814 by J. M. Hermann, a Bavarian 
engineer. A polar planimeter, which is a development of the 
planimeter invented by Jacob Amsler at Konigsberg in 1854, is 
the one most extensively used. By it the area of any figure is 
obtained by going around the boundary line of the figure with 
a tracing point and noting the numbers that are indicated on a 
measuring wheel when the operation of tracing begins and ends. 

Integrators and integraphs also serve for the measurement of 
areas; they are adapted, moreover, for making far greater compu- 
tations and solving more complicated problems, such as the calcu- 
lation of moments of inertia, centres of gravity, etc. The integraph 
(see Art. 170, Notes 2, 3) is the superior instrument, for it directly 



196.] PLANIMETERS, INTEGRAPRS. 349 

and automatically draws the successive integral curves. These 
give a graphic representation of the integration, and are of great 
service, especially to naval architects. The measure of an ordi- 
nate of the first integral curve, when multiplied by a constant 
belonging to the instrument, gives a certain area associated with 
that ordinate (see Art. 170). 

Note 1. A bicycle with a cyclometer attached may be regarded as a 
mechanical integrator of a certain kind ; for by means of a self-recording 
apparatus it gives the length of the path passed over by the bicycle. 

Note 2. Planimeters and integrators are simple, and it is easy to learn 
to use them. 

Note 3. A brief account of the planimeter, references to the literature on 
the subject, and a note on the fundamental theory, will be found in Murray, 
Integral Calculus, Art. 88, and Appendix, Note F. Also see Lamb, Cal- 
culus, Art. 102 ; Gibson, Calculus, § 130. For a fuller account see Henrici, 
Report on Planimeters (Report of Brit. Assoc, for Advancement of Science, 
1891, pages 496-523) ; Hele Shaw, Mechanical Integrators (Proc. Institution 
of Civil Engineers, Vol. 82, 1885, pages 75-143). For references concerning 
the integraph see Art. 170, Note 3. 

N.B. Interesting information concerning planimeters, integrators, and 
the integraph, with good cuts and descriptions, are given in the catalogues of 
dealers in drawing materials and surveying instruments. 

Note 4. For approximate integration by means of series see Art. 199. 



CHAPTER XXIII. 

INTEGRATION OF INFINITE SERIES. 

197. Integration of infinite series term by term. It is beyond 
the limits of a short course in calculus to investigate the condi- 
tions under which an infinite series can properly be integrated 
term by term ; in other words, to determine what conditions 
must be satisfied in order that equation (3) Art. 143 (e) may be 
true.* 

It must suffice here merely to state the theorem that applies 
to most of the series that are ordinarily met in elementary mathe- 
matics ; viz. : 

A power series (Art. 145) can be integrated term by term through- 
out any interval contained in the interval of convergence and not 
reaching out to the extremities of this interval. (For proof see 
Osgood, Infinite Series, Art. 40.) The next two articles give 
applications of this theorem. 

198. Expansions obtained by integration of known series. Three 
important examples of the development of functions into infinite 
series by the aid of integration will now be given. 

The three expansions for tan -1 x, sin -1 #, log (1 -f- x), in Exs. 1, 2, 
3, can also be derived by means of Maclaurin's theorem. (See 
Art. 152, Ex. 10 (3).) 

EXAMPLES. 

Ex. 1. For -1<s<1 

-J— = 1-8* + **-.». (1) 

1 + X 1 

:. p-^-= ( X dx- ( x x*dx+ Cx^dx , (Art. 197) 

Jo 1 + x 2 Jo Jo Jo 

i.e. t2LK 1 X = K-~+- r . (2) 

3 5 

* See Art. 147 and Infinitesimal Calculus, Arts. 172, 173. 
350 



197, 198.] INTEGRATION OF INFINITE SERIES. 351 

This is known as Gregory's series.* (For complete generality the term 
± mr, (n = 0, 1,2, •••), should be in the second member.) Series (1) oscil- 
lates when x = 1 ; but by a theorem on series (see Chrystal, Algebra, 
Vol. II., Chap. XXVI., § 20) series (2) is convergent and represents tan -1 x 
even when x = 1. 

Note 1. Series (2) can be used to calculate ir. On putting x = 1 (2), 
there is obtained 

/ K I -, 1,1 1 , 

This is a very slowly convergent series. More rapidly convergent series 
for calculating ir are the following : 

(6) - = 4 tan" 1 - - tan" 1 — ; (Machin's Series t) 
4 5 239 

(c) - = tan" 1 - + tan -1 - • (Euler's Series 1) 

Exercises. Show by elementary trigonometry that formulas (b) and (c) 
are true. Compute the value of ir correctly to four places of decimals: 
(1) by using formula (6) and Gregory's series ; (2) by using formula (c) 
and Gregory's series. (The correct value of ir to ten places of decimals is 
3.1415926536.) 

Ex. 2. For - 1 < x < 1 

_J_ = (1 _ x ,yi = , + i x2 + LlS^ + lll^e + .... 

On integrating between the end values and 1, as in Ex. 1, there results 

This series is due to Newton, and was used by him in computing the value 
of 7r. When x = \ this series gives 

7T 1 1 1.3 1.3-5 



6 2 2- 3- 2 3 2- 4- 5- 2 5 2- 4- 6. 7- 2 7 

Exercise. Using the last result calculate ir correctly to four places of 
decimals. 



* Discovered in 1670 by James Gregory (1638-1675), professor of mathe- 
matics at St. Andrews and later at Edinburgh. It was also found by Leibnitz 
(1646-1716). This series can also be derived independently of the calculus 
(see texts on Analytical Trigonometry). 

t John Machin, died 1751, was professor of astronomy at Gresham College, 
London. % Leonhard Euler, 1707-1783. 



352 INTEGRAL CALCULUS. [Ch. XXIII. 

Note 2. For historical information concerning trigonometry and the 
computation of w, see Murray, Plane Trigonometry, Appendix, Note A, and 
Note C (Art. 6) ; Hobson, article " Trigonometry" (Ency. Brit., 9th edition); 
also article "Squaring the Circle" {Ency. Brit., 9th edition). 

Ex. 3. For - 1<»<1 

-J— = 1 - x + x*- x s + .... (1) 

1 + x 

On integrating between the end values and x, as in Exs. 1, 2, there results 
log(l + x)= oc -\^ + |ar* -\ntfi + .... (2) 

This is called the logarithmic series.* {Here the base is e.) 

The members of (2) are equal for values of x as near 1 as one pleases. It 
is also easily shown that they are finite and continuous for x = 1. Accord- 
ingly, formula (2) is true also when x = 1. 

On putting x = 1 in (2), log 2 = 1 - § + \ — \ + ..., a very slowly conver- 
gent series. 

On putting x=-l in (2), logO=- (1 + J + i + * + ...) = _ oo. (See 
Art. 145.) 

Note 3. Except for small values of x series (2) is very slowly convergent. 
A more rapidly convergent, and thus more useful, serits for the computation 
of logarithms can be derived from (2) , as follows. On putting — x for x in (2), 

log(l -x) = -x-lx*r-^x*-%x i . (3) 

••• log^ = 2 {x + i x* + \* + -..). (4) 

1 — x 

On substituting for x this becomes 

2 m + 1 

l 0g ™+l =2 r-J-- + 1 -+ 1 -+...1. (5) 

m L2m + 1 3(2»rc+l) 8 5(2m + l) 6 J 



Hm = l, log 2 = 2(| + g^ + J^ +•••) = . 693. 

If m = 2, log 3 - log 2 = 2 (1 + ^L + -i^ + ...) = .406. 

.-.log 3= 1.099. 

Exercises. (1) Find log 4 to base e, by putting m = 3 in (5), assuming 
the value of log 3. (2) Find the logarithms (to base e) of 5, 6, 7, 8, 9, 10, in 
a similar way. (The logarithms of 4, 5, 6, 7, 8, 9, 10, to base e, to three 
places of decimals, are respectively 1.386, 1.609, 1.792. 1.946, 2.079, 2.197, 
2.303.) 

* Apparently first obtained in 1668 by Nicolaus Mercator of Holstein. 



198, 199.] INTEGRATION OF INFINITE SERIES. 353 

199. Approximate integration by means of series. The methods 
described or referred to in Arts. 194-196 for evaluating a definite 
integral 

^f(x)dx (1) 



£ 



yield a numerical result only. They do not give any information 

as to the anti-differential of f(x) dx. 

Some information, however, about the anti-differential of f(x) dx 
can be obtained in certain cases (see Art. 197) by expanding /(#) 
in a series in ascending or descending powers in x and then inte- 
grating this series term by term. The new series thus obtained 
represents the anti-differential of f(x)dx for values of x in some 
particular interval of convergence. From this series an approxi- 
mate value of (1) can be obtained, if the end-values a and b are 
in the interval of convergence. 

Instances have been given in Art. 198, thus 

• -I? 1 r 1 dx -,1,1 

mEx.l, _^_ = 1 + . 

Jo 1 + x 2 3 5 

* dx 1 . 1 1-3 



• tj, Q p dx 1 . 1 

in Ex. 2, I — =--\ 

Jo VT^-2 2^2- 3- 2 



! 



VI^s 2 2 • 3 • 2 3 2 • 4 • 5 • 2 s 



EXAMPLES. 

1. Given that e x = 1 + x+ — + — + ••• (Art. 152, Ex. 7), show that 
e x dx = e x + c, in which c is a constant. 

2. Given that cos £ = 1 - — + , and that sin x = x - — + 

2! 4! 3! 5! 

(Art. 152, Exs. 2, 5), show that I cos x dx = sin x + c, and that i sinxdx 

= — cos X + c. 

3. Find an approximate value of the area of the four-cusped hypocycloid 
inscribed in a circle of radius 8 inches. (This area can also be found exactly. 
See Art. 209, Note 5, Ex. 1.) 

4. Eind an approximate value of the length of the ellipse x = a sin 0, 
y = & cos0. [Here <f> is the complement of the eccentric angle for the point 



354 INTEGRAL CALCULUS. [Ch. XXIII. 

It will be found (Art. 209) that 

IT 

length s = 4af Vl - e 2 sin 2 <p d<p. {a) 

On expanding the radical by the binomial theorem and taking the term 
by term integral of the resulting convergent series it will be found that 

— C 1 - (l) 2 f - (H) 2 f - (Hrl) 2 ! - -]- ?> 

5. Apply result (b) of Ex. 6 to find the length of the ellipse whose semi- 
axes are 5 and 4. (To three places of decimals.) 

6. The time of a complete oscillation of a simple pendulum of length 7, 
oscillating through an angle oc(<7r) on each side of the vertical, is 



M 



Vl — k l sin 2 
Show that this time 



( ^ , in which k = sin \ a. (c) 



= 2 



Note 4. Integrals (c) and (a) in Exs. 6 and 4 are known respectively as 
" elliptic integrals of the first and the second kind." The symbols F(k, 0), 
E(e, 0) are usually employed to denote these integrals (the upper end-value 
here being 0). Knowledge of these integrals was specially advanced by 
Adrien Marie Legendre (1752-1833). See Art. 192, Note 4. 



7. Show that: 



(1) P-* 

< 2 > Jo' 



dx =1 , 1 1.1-3 1 1-3-5 J_ 
4 2*52- 5*92- 4- 6' 13 



x* 

dx _i_l 2 1 2jjj JL_JL 2-5-8 _1 
V (i X x*) 2 ~ 4 3 7 ' 1 - 2 ' 3 2 10 ' 1 • 2 • 3 ' 3 3 



(3) £■* 



dx_ =1 \ 1 1_ lj_4 1 _! !- 4 - 7 . I + 
^s 6*3 11 ' 1.2* 3 2 16* 1-2-3 ' 3 3 



CHAPTER XXIV. 

SUCCESSIVE INTEGRATION. MULTIPLE INTEGRALS. 
APPLICATIONS. 

200= In Chapter VI. (see Arts. 68, 69, 70), successive deriva- 
tives and differentials of functions of a single variable were 
obtained. In Chapter VIII. (see Arts. 79, 80, 82), successive par- 
tial derivatives and partial differentials of functions of several 
variables were discussed. In this chapter processes which are the 
reverse of the above are performed and are employed in practical 
applications. 

201. Successive integration : One variable. Applications. 

Suppose that J f(x)dx =f 1 (x), (1) 

fMx)dx = f 2 (x), (2) 

ff 2 (x)dx=f s (x). (3) 

Then, by (3) and (2), /,(*) =f[ffi(*) dx~\ dx ; (4) 

By (4) and (1), f s (x) = jT ff f f(x)dxj dx dx. (5) 

This is written f s (x) = f f C f(x)(dxf, 

or, more usually, f 3 (x) = f f ff(x) dx 3 . (6) 

The second member of (6) is called a triple integral. Similarly, 

the second member in (4) is usually written j | f x (x) dx 2 , and is 
called a double integral. 

In general, J I J ••• J f(x)dx n denotes the result obtained by 

355 



356 INTEGRAL CALCULUS. [Ch. XXIV. 

integrating f(x)dx n times in succession. This integral is indefi- 
nite unless end values of the variable be assigned for each of the 
successive integrations. This integral and the integrals in (4) and 
(5) are called multiple integrals. 

Note. It should be observed that here dx n denotes dx dx dx ••• to n factors, 
i.e. (dx) n , and not d • x n (i.e. nx n ~ 1 dx). [Compare Art. 70.] 

EXAMPLES. 
1. Find 



CC (x' 2 dxs= CI C[ Cx^dxldxldx 



= %- + fax* + c 2 x + c 3 ; 
oU 

for, since Ci is an arbitrary constant, ^ may be denoted by an arbitrary con- 
stant k\. 

d?y 

3. Determine the curves for every point of which — \ = 0. Which of 

these curves goes through the points (1, 2), (0, 3) ? Which of these curves 
has the slope 2 at the point (3, 5) ? 

On integrating, -^ = c\. 

On integrating again, y = C\X + c 2 , 

which represents all straight lines. 

" For the line going through (1, 2) and (0, 3), 2 = Ci 4- c 2 and 3 = + c 2 ; 
whence Ci = — 1, c 2 = 3. Hence the line is x + y = 3. 

For the line having the slope 2 at (3, 5), C\ = 2 and 5 = 3 ci + c 2 , whence 
c 2 = — 1. Hence the line is y = 2 x — 1. 

4. Determine the curves for every point of which the second derivative 
of the ordinate with respect to the abscissa is 6. Which of these curves 
goes through the points (1, 2), (— 3, 4) ? Which of them has the slope 3 at 
the point (-2,4)? 



201, 202.] SUCCESSIVE INTEGRATION. 357 

ST.Bo The student is recommended to write sets of data like those in 
Exs. 3-7, and determine the particular curves that satisfy them. He is also 
recommended to draw the curves appearing in these examples. 

5. Determine the curves for every point of which the second deriva- 
tive of the ordinate with respect to the abscissa is 6 times the number of 
units in the abscissa. Which of these curves goes through the points (0, 0) 
(1, 2) ? Which of them has the slope 2 at (1, 4) ? 

d 2 u 

6. Determine the curves in which the second derivatives -=-| from point 

to point vary as the abscissas. Find the equation of that one of these curves 
which passes through (0, 0), (1, 2), (2, 5). Find the equation of that one of 
these curves which, passes through (1, 1), and has the slope 2 at the point 
(2, 4). 

7. Determine the curves in which the second derivative of the abscissa 
with respect to the ordinate varies as the ordinate. Which of these curves 
passes through (0, 1), (2, 0), (3, 5) ? Which of them has the slope J at 
(1, 2), and passes through (— 1, 3) ? 

8. A body is projected vertically upward with an initial velocity of 1000 
feet per second. Neglecting the resistance of the air and taking the accelera- 
tion due to gravitation as 32.2 feet per second, calculate the height to which 
the body will rise, and the time until it again reaches the ground. 

9. Do Ex. 20, Art. 68. 

10. When the brakes are put on a train, its velocity suffers a constant 
retardation. It is found that when a certain train is running 30 miles an 
hour the brakes will bring it to a dead stop in 2 minutes. If the train is to 
stop at a station, at what distance from the station should the engineer 
whistle "down brakes" ? (Byerly, Problems in Differential Calculus.) 

202. Successive integration : several variables. Suppose that 

jf(®, y> z ) dz =M ai > v> z )> CO 

J/iO», y, z) dy = f 2 (x, y, z), (2) 

J/afa y, z) dx =f s (x, y, z). (3) 

The integration indicated in (1) is performed as if y and x were 
constant; the integration in (2) as if x and z were constant; the 
integration in (3) as if z and y were constant. (Compare Arts. 79, 
80.) 



358 INTEGRAL CALCULUS. [Ch. XXIV. 

From (3) and (2), f 3 (x, y, z) =j j J/^, y, z) dy}dx; (4) 

from (4) and (1), = J \ f[ff(& V, *) <**]dy \ dx. (5) 



The second member in (4) is often written 

JJfi(x,y,z)dydx- 9 (6) 

the second member in (5) is often written 

f(x, y, z)dzdydx. (7) 



fffi 



The integral in (6) is called a double integral, and the integral 
in (7) a triple integral. 

Note 1. It should be observed that according to (2), (3), and (4), inte- 
gral (6) is obtained by first integrating fi(x, y, z) with respect to y, and then 
integrating the result with respect to x ; in (7), according to (1), (2), (3), 
and (5), the first integration is to be made with respect to z, the second with 
respect to y, and the third with respect to x. That is, the first integration sign 
on the right is taken with the first differential on the left, the second integra- 
tion sign from the right with the second differential from the left, and so on. 
When end-values are assigned to the variables, careful attention must be paid 
to the order in which the successive integrations are performed. 

Note 2. The notation used above for indicating the order of the variables 
with respect to which the successive integrations are to be performed, is not 
universally adopted. Oftentimes, as may be seen by examining various texts 
on calculus and works which contain applications of the calculus, integrals (6) 
and (7) are written 

( J7i(*i ?i «)^^) ( ( I /(#» V, z) dx dy dz respectively. 

In this notation the first integration sign on the right belongs to the first 
differential on the right, the second integration sign from the right to the 
second differential from the right, and so on ; and the integrations are to be 
made, first with respect to z, then with respect to y, and then with respect to x. 
In particular instances, the context will show what notation is employed. 



EXAMPLES. 



J" S $ x2yz * dz d y dx =§ ^ x Hi [— + oi) dy dx 



-W« 



M(7^H + ^ + *)(T + " 



202, 203.] SUCCESSIVE INTEGRATION. 

/*4 (**> /*3 /*x=4 fy=2 /*2=3 

2. ( I ( x 2 yz s dz dy dx (i.e. I J ^ I x 2 yz* dz dy dx) 

= ££* [i + c ]> dx = f XT* 2 * c ^ to 

4 J2 L2 Ji 2 4J2 

= I P a: 3 (x 6 - x 3 ) <fe = 28£fr. 
4. Evaluate the following integrals : (1) | | I xy 2 z dz dy dx. 



359 



(2) f 3a f ° (3 w -2v)dw dv. 
2 

fir faCl-cosB) „ , , 

(4) j | r 2 cos Odrdd, 

n 

W X X io ' Sm "*" 

< 8 > I. X 



^ X X 



Vsi - t 2 ds dt. 



dd. 



rdr dd. 



("la /* c o s-1 (<W , -. 

(7) | j, W r»*. 

7T 

(9) || Va 2 - r 2 • rdr <Z0. 



203. Application of successive integration to finding areas : rec- 
tangular coordinates. 

EXAMPLES. 

1. Find the area between the curve y 2 = 8 x, the x-axis, and the ordinate 
for which x = 3. 

At P, any point within the figure OWM 
whose area is required, suppose that a rectan- 
gle PQ having infinitesimal sides dx and dy 
parallel to the axis is constructed. The area 
WM is the limit of the sum of all rectangles 
such as PQ which can be constructed side by- 
side in WM. Let one of the vertical sides of 
j^ — ^ the rectangle be produced both ways until it 
meets the curve and the x-axis in T and S; 
complete the rectangle TV as in the figure. 
First, find the area of the rectangular strip TV by finding the limit of the 
sum of the rectangles PQ inscribed in it from S to T; then find the limit 
of the sum of the strips like TV which can be inserted between OY and MW. 




Fig. 122. 



3G0 INTEGRAL CALCULUS. [Ch. XXIV. 

Area TV = lim > (rectangles P$) = J dy dx = VSx dx. (1) 



y at 5 

as at if - „ = v/g^ 

Area OJOF = lim ^ (strips IT) = j^[ j^ <fy] <*» (2) 

x at O 

= 2 V2 i x 2 dx = 4 V6 square units. 

The last expression in (2) is usually written 4 4 8a= <^|/ dx. 

The area of WM may also be found by finding the limit of the sum of the 
rectangles PQ which may be inserted between B and U, and then finding 
the limit of the sum of the strips like BL which may be inserted between 
OM and W. Thus, 

area BL = fj*** dx dy = £ dx dy = (s~ £) dy ; (3) 

area OMP = j^/( 8 -f ) * = £*(« - J) * = 4V6. (4) 

From (3) and (4), area OMP = \ \ dxdy. 

8 

Note 1. The last expression in (1) is y dx, the element of area employed 
in Art. 166. 

Note 2. Ex. 1 has been solved as above merely in order to give a prac- 
tical application of double integration. 

Note 3. For finding areas by double integration in the case of polar 
coordinates, see Art. 208, Note 3. 

2. Express some of the areas in Art. 181 by double integrals, and per- 
form the integrations. 

3. Find by double integration the area included between the parabolas 
3 y 2 = 25 x and 5 x 2 = 9 y. [See Murray, Integral Calculus, Art. 61, Ex. 1.] 

204. Application of successive integration to finding volumes: 
rectangular coordinates. 

EXAMPLES. 

1. Find the volume bounded by the surface whose equation is 

y* , rf 

a 2 "*" b 2 + c 2 

Fig. O-ABC represents one-eighth of the volume required. Suppose that 
an infinitesimal parallelopiped P\Qz is taken at Pi(x, y, o), having infinitesi- 



203, 204.} 



SUCCESSIVE IXTEGBATION. 



361 



mal sides dx, dy, dz, parallel to the x-, y-, and z-axes, respectively. The 
volume of O-ABC is the limit of the sum of all infinitesimal parallelopipeds 
such as Pi Qz which can be enclosed by OB A, OAC, OCB, and the curvi- 




Fig. 123. 



linear surface ABC. Construct a parallelopiped PQi by producing the 
vertical faces of P x Qs to the height Pi P. (The point P(x, y, z) is taken on 
the surface ABC.) 

Vol. PQ 1 = \ dzdydx = \ \ a2 b2 

J 2 atP x " \_Jz=S) 



dz 



dy dx. 



(1) 



Note 1. The numbers x and y are constant along P X P, and, accordingly, 
in the integration of (1) x and y are treated as constants. 

Xow take a slice BGL the planes of whose faces coincide with two faces 
of P$i, as shown in the figure. 

Vol. slice BPGLS = limit of sum of parallelopipeds PQ 1 from S to G. 



VX 2 V 2 
I *_ 

That is, vol. slice EG = \ \ a2 b2 dz 

Jy at S Jz=Q 



dx 



={c^[r Vi ~^-H- 



(2) 



Note 2. The number x is constant along SG, and, accordingly, in the 
integration of (2) x is treated as a constant. 

Xow find the limit of the sum of all infinitesimal slices like BGL from 
OCB to A ; i.e. from x = to x = a. This limit is the volume of O-ABC. 



362 INTEGRAL CALCULUS. [Ch. XXIV. 



vol. 0-ABC=\_ } | _ "| | •• >'dz\dy}dx 




br^] 



« 2 b2 dzdydx. (3) 

On performing the integrations indicated in (3) (see Ex. 4 (5), Art. 202), it 
will be found that 

vol. O-ABC = 1 7T abc. Hence vol. ellipsoid = f w a6c. 
Note 3. Result (3) may be written f * at A C y&iG C z at p ^ ^ ^ 

JxatO JyatS Jz at Px 

Note 4. The initial element of volume P x Q s , i.e. dx dy dz, is an infinitesi- 
mal of the third order; the parallelopiped PQ X is an infinitesimal of the 
second order ; the slice BGL is an infinitesimal of the first order. 

Note 5. Equally well, slices may be taken which are parallel to the 
xz-plane or to the z/s-plane. 

Note 6. Instead of the parallelopiped PQ U equally well, a similar paral- 
lelopiped can be taken whose finite edges are parallel to the y-axis, or to the 
a-axis. 

2. Perform the integrations indicated in Ex. 1. 

3. Do Ex. 1 by taking the elements in the ways indicated in Notes 5 
and 6. 

4. From the result in Ex. 1 deduce the volume of a sphere of radius 
a. Also deduce the volume of this sphere by the method used in Ex. 1. 
(Compare with the methods used in Art. 182, Ex. 19 and Note 3.) 

5. Two cuts are made across a circular cylindrical log which is 20 inches 
in diameter ; one cut is at right angles to the axis of the cylinder, the 
other cut makes an angle of 60° with the first cut, and both cuts intersect 
the axis of the cylinder at the same point. Find the volume of each of the 
wedges thus obtained. 

6. As in Ex. 5, for the general case in which the radius of the log is a 
and the angle between the cuts is a. Thence deduce the result in Ex. 5. 

7. The centre of a sphere of radius a is on the surface of a right cyl- 
inder the radius of whose base is -. Find the volume of the part of the 
cylinder intercepted by the sphere. 

8. Taking the same conditions as in Exs. 5, 6, excepting that the cuts 
intersect on the surface of the log, find the volume intercepted between the 
cuts. 



204, 205.] 



8 UCCESSIVE INTEGRA TION. 



363 



205. Application of successive integration to finding volumes ; 
polar coordinates. 

A. The use of polar coordinates in rinding volumes sometimes 
leads to easier integrations than does the use of rectangular 

coordinates. 

Let 0, the origin, be taken 
as pole. The infinitesimal ele- 
ment of volume is formed 
as follows : Take any point 
P(r, 0, 4>). [Here r = OP,0 = 
angle POZ, cf> = angle XOM, 
OM being the projection of OP 
on XOY. In other words, 
<£ = the angle between the 
plane XOZ and the vertical 
plane in which OP lies.] Pro- 
duce OP an infinitesimal dis- 
tance dr to P 1} and revolve 
OPP x through an infinitesimal 
angle dO in the plane ZOP to the position OQ. Now revolve 
OPP x Q about OZ through an infinitesimal angle dcj>, keeping 
constant. The solid PP X QR is thus generated. Its edges 
PPj, PQ, PR are respectively dr, rdO, rsmOdcf>; its volume (to 
within an infinitesimal of an order lower than the third) is 
r 2 sin dr d<f> dO. On determining the proper limits for r, <£, 0, and 
integrating, the volume required is obtained. 

Ex. 1. Find the volume of a sphere of radius a, using polar coordinates 
and taking on the surface of the sphere and OZ on the diameter through O. 
(It will be found that the volume is given by the integral in Art. 202, Ex. 4, 
(6). See Murray, Integral Calculus, Art. 63, Ex. 1.) 

B. The element of volume can be chosen in another way, which 
sometimes leads to simpler integrations than are otherwise obtain- 
able. An instance is given in Ex. 2 below. 




Fig. 124. 



EXAMPLES. 

2. Another way of doing Ex. 7, Art. 204. 

In the figure, O-ABC is one-eighth the sphere, and the solid bounded by 
the plane faces ALBO, AKO, the spherical face ALBVA, and the cylindrical 



364 



INTEGRAL CALCULUS. 



[Ch. XXIV 



face AVBOKA is one-fourth of the part of the cylinder intercepted by the 
sphere. 

In AOK take any point P. 
Let OP = r, and angle AOP = 9. 
Produce OP an infinitesimal dis- 
tance dr to Pi, and revolve OPPi 
through an infinitesimal angle dd. 
Then PP± generates a figure, two 
of whose sides are dr and rdd. 
Its area (to within an infinitesimal 
of an order lower than the second) 
is r dr dd. (See Art. 208, Note 3, 
Ex. 8.) 

On this infinitesimal area as 
a base, erect a vertical column 
to meet the sphere in M. Then 
Va? — r 2 , and the volume 
rdrdd. 




PM. 

of the column is Va' 2 — r' 2 
This is taken as the element of 
volume ; the limit of the sum of these columns standing on AOK is the vol- 
ume required. Keeping 6 constant, first find the limit of the sum of the 
columns standing on the sector extending from to K whose angle is dd. 

/*j*=acos0 , 

Since OK = a cos d, this limit is \ Va? — r 2 • rdrdd. This gives the 

Jr=0 

volume of a wedge-shaped slice whose thin edge is OB. One-fourth of the 
volume required is the limit of the sum of all the wedge-shaped slices of this 
kind that can be inserted between AOB and COB; that is, from d = to 



Fig. 125. 



vol. required = 41 2 \ 
Je=o Jr= 



Va 2 - r 2 .rdrdd = f 7ra 3 - fa 3 . 

[See Art. 202, Ex. 4 (9).] 

In this instance this is a very much shorter way of deriving the volume 
than by starting with the element dx dy dz. as in Art. 204. 

3. Find the volume of a sphere of radius a, taking at the centre: 
(1) choosing the element of volume as in A ; (2) choosing it as in B. 

4. The axis of a right circular cylinder of radius b passes through the 
centre of a sphere of radius a {a > b). Find the volume of that portion of 
the sphere which is external to the cylinder. 



CHAPTER XXV. 

FURTHER GEOMETRICAL APPLICATIONS OF 
INTEGRATION. 

206. In this chapter the calculus is used for finding volumes 
in a particular case, for finding areas of curves whose equations 
are given in polar coordinates, for finding the lengths of curves 
whose equations are given either in rectangular or in polar coordi- 
nates, for finding the areas of surfaces in two special cases, and 
for finding mean values of variable quantities. 

If.B. Many of the problems in this chapter are presented in a general 
form. In such cases the student is recommended, when he obtains the 
general result, to make immediate application of it to particular concrete 
cases. 

207. Volumes of solids the areas of whose cross-sections can be 
expressed in terms of one variable. In Art. 182 the volumes of 
solids of revolution were found by making cross-sections of the 
solid at right angles to the axis of revolution, taking these cross- 
sections an infinitesimal distance apart, and finding the limit of 
the sum of the infinitesimal slices into which the solid is thus 
divided. This method of finding the volume of a solid can some- 
times be easily applied in the case of solids which are not solids 
of revolution. The general method is : (a) to take a cross-section 
in some convenient way ; (b) to express the area of this cross- 
section in terms of some variable ; (c) to take a parallel cross-sec- 
tion at an infinitesimal distance from the first cross-section ; (d) to 
express the volume of the infinitesimal slice thus formed, in terms 
of the variable used in (b) : (e) to find the limit of the sum of the 
infinite number of like parallel slices into which the solid can 
thus be divided. There is often occasion for the exercise of judg- 
ment in taking the cross-sections conveniently. 

865 



366 



INTEGRAL CALCULUS. 



[Ch. XXV. 



EXAMPLES. 

1. Find the volume of a right conoid with a circular base of radius a and 
an altitude h. 

Note 1. A conoid is a surface which may be generated by a straight line 
which moves in such a manner as to intersect a given straight line and a given 
curve and always be parallel to a 
given plane. In the conoid in this 
example the given plane is at right 
angles to the given straight line, and 
the perpendicular erected at the 
centre of the circle to the plane of 
the base intersects the given straight 
line. 

Let LM be the fixed line and ARB 
the fixed circle having its centre at 
C. Take a cross-section PQR at 
right angles to LM, and, accordingly, 
at right angles to a diameter AB. 
Let it intersect AB in D, and denote 
CD by x. 
Area 

PQR = ±PD-QR = PD- QD. 

Now PD = h, and, by elementary geometry, 




Fig. 126. 



QD 
area PQR 



VAD ■ DB = V(a - x) (a + x) = Va 2 - x 2 . 



h Va' 2 - x 2 . 

Now take a cross-section parallel to PQR at an infinitesimal distance from 
it. Since CD has been denoted by », this infinitesimal distance may be 
denoted by dx. 

Vol. LM-BQARB = 2 vol. LG-TSAT 

xzXA 

= 2 lim (sum of slices PQR) 



= 2 \f 



Va 2 — x 1 dx 



-a 2 h. 



That is, the volume of the conoid is one-half the volume of a cylinder of 
radius a and height h. (See Echols, Calculus, Ex. 3, p. 266.) 

Note 2. As already observed, finding the volumes of solids of revolution 
is a special case under this article. 

Note 3. Two general methods of finding volumes have now been shown, 
namely, the method shown in Arts. 204, 205, and the method shown in this 
article. 



207, 208.] 



ABEAS: POLAR COORDINATES. 



367 



2. Do Ex. 1, denoting AD by x. 

3. Do Ex. 8, Art. 182 and Ex. 1, Art. 204 by method of this article. 

4. Find the volume of a right conoid of height 8 which has an elliptic 
base having semi-axes 6 and 4, and in which the fixed line is parallel to the 
major axis. Eind the volume in the general case in which the height is /i, 
the semi-major axis a, and the semi- minor axis b. 

5. A rectangle moves from a fixed point, one side varying as the dis- 
tance from the point, and the other side as the square of this distance. At 
the distance of 3 feet the rectangle is a square whose side is 5 feet. What 
is the volume generated when the rectangle moves from the distance 2 feet 
to the distance 4 feet ? 

6. On the double ordinates of the ellipse b 2 x 2 + a 2 y 2 = a 2 b 2 , and in planes 
perpendicular to that of the ellipse, isosceles triangles having vertical angles 
2 a are erected. Eind the volume of the surface thus generated. 

7. A circle of radius a moves with its centre on the circumference of an 
equal circle, and keeps parallel to a given plane which is perpendicular to the 
plane of the given circle : find the volume of the solid thus generated. 

8. Two cylinders of equal altitude h have a circle of radius a for their 
common upper base. Their lower bases are tangent to each other. Eind the 
volume common to the two cylinders. 

208. Areas: polar coordinates. Suppose there is required the 
area of the figure bounded by the curve whose equation is 
f(r, 0) = 0, and the radii vectores drawn to two assigned points 

on this curve. 



<^r 2 ,0 2 ) 




Let LG be the curve 
f(r, 0) = O, and P and 
Q the points (r 1? 0j) 
and (r 2 , 2 ) respectively; 
it is required to find 
the area POQ. Sup- 
pose that the angle POQ 
is divided into n equal 
angles each equal to AO, 
and let VOW be one of 
these angles. Denote Fas 
the point (r, 9). Through 
V, about O as a centre, 



draw a circular arc intersecting O W in M. 



368 INTEGRAL CALCULUS. [Ch. XXV. 

Through W, about as a centre, draw a circular arc intersecting 
OV in JV. Denote MWhy Ar. 

Then, area OVM = ±r 2 A0 (PL Trig., p. 175), and area ONW 
= i(r + Ar) 2 A<9. 

Let "inner" and "outer" circular sectors, like VOM and NO W 
in the case of VW, be formed for each of the arcs like VW which 
are subtended by angles equal to A0 and lie between P and Q. It 
is evident that 

total area of inner sectors <area POQ< total area of outer sectors. 

(i) 

In the case of the arc VW the difference between the inner and 
outer sectors is VMWN. On noting this difference for each arc 
and transferring it to the radius vector OPS, as indicated in the 
figure, it is apparent that the total difference between the areas 
of the inner and outer sectors is PBCS. Now 

area PBCS = area OSC - area OPB = i (OS 2 - OP 2 ) AO ; 

and this approaches zero when A0 approaches zero. 

From these facts and relation (1) it follows that 

Area POQ = limit of area of inner sectors (or outer sectors) 
when A0 approaches zero, that is, when the number of these 
sectors becomes infinitely great. That is, 

Area POQ = limit of sum of areas of sectors VOM from OP 
to OQ when A0 approaches zero 

= lim A ^ X i r ^° = 2 f ^ 2 dO. (See Art. 166. ) 

Note 1. The element of area in polar coordinates is thus \r' l d9\ this is 
the area of an infinitesimal circular sector, of which the radius is r and the 
angle is an infinitesimal, dd. The differential of the area also has the same 
form 1 r 2 dd. In the element of area dB must be infinitesimal, in the differen- 
tial dd need not be infinitesimal. (See Art. 67 b.) 

Note 2. It is not necessary that the angles A0 be all equal. (See Art. 166, 
Note 3.) 



208.] 



AREAS: POLAR COORDINATES. 



369 



EXAMPLES, 

1. Find the area of a loop of the curve r = a sin 2 0. 

It is first necessary to find the values of at the beginning and at the end 
of a loop. At (see Fig., page 464) r = ; hence, sin20 = O at 0. If 



sin 2 = 0, then 



0, 7r, 2 7r, •••, and, accordingly, = 0, 



Any pair of consecutive values, say and -, are values of at at the 
beginning and end of a loop. 

.-. area of a loop = 1 1 ^(20 = 5L j sin 2 2 = — I | (1 - cos 4 

4 L 4 Jo 8 

2. Find the area of one of the loops of the curve r 



a sin 3 0. 

3. Find (1) the area of a loop of the lemniscate r 2 = a 2 cos 2 ; (2) the 
area of a loop of the curve r 2 = a 2 cos nd. 

4. Show that (1) the area included between the hyperbolic spiral rd = a 
and any two radii vectores is proportional to the difference between the 
lengths of these radii vectores ; (2) the area included between the logarithmic 
spiral r = e ad and any two radii vectores is proportional to the difference 
between the squares on these radii vectores. 

5. Find the area enclosed by* the cardioid r 2 = a 2 " cos — 

J 2 

6. Find the area of the oval r = 3 + 2 cos 0. 

7. Compute the area of the loop of the folium of Descartes x s + y z = 3 a xy. 

Suggestion for Ex. 7 : Change to polar coordinates, and then use the 
substitution z = tan 0. 

Note 3. On finding areas of curves by double integration. For the sake 
of illustration an example will be shown in which areas, in polar coordinates, 
are found by double integration. 

8. Find the area of the circle 
r = 2 a cos 0. 

Take any point P in ODA. 
Let OP- r, angle AOP=6. Pro- 
duce OFa distance Arto Q ; revolve 
OPQ through an angle A0. Then 
PQ sweeps over the area PQJRS. 

Area PQRS 
= \ OQ 2 • A0 - i OP 1 • A0 
= r • Ar • A0 + \ (Ar) 2 • A0. 




Fig. 128. 



370 INTEGRAL CALCULUS. [Ch. XXV. 

One can proceed to find the limit of the sum of the areas like PQBS in 
ODA, in either of the two following ways (a) and (5). 

(a) Starting with PQBS as an element of area, find the area of the 
sector BOC; then, using BOC as an element of area, derive therefrom 
the area of ODA. Thus, 

r=OB 

X~^ f » — 2 a cos 

area BOC = lim A ^o ^ -P#-R# = I _ rdr-AO; 



r=0 
e=AOT 



area OZL4 = lim A fe ^ BOC = ( * C 



2aCOs6 rdrd6=^ 

2 



(&) Starting with PQBS as an element of area, find' the area of the 
circular strip GDF ; then using GDF as an element of area, derive there- 
from the area of ODA. Thus, 



=cos 

OA 



H — ) 

x-\ 0A re=cos- 1 (~^ 

area GDF = lim A 0=o Z, PQRS = J ' y2a 'rdd-Ar 



0=0 



area ODA = lim Ar :=o2y ^^ = f " J 



cos _1 ( ^- ) 



<dd dr 

r=0 

.-. area of circle = 2 area ODA = ira\ [Ex. 4 (7), Art. 202.] 

In this method of computing areas the infinitesimal element of area is 
thus rdrdd. 

Note 4. For discussions on the sign to be given to an area, on the areas 
of closed curves, and on the area swept over by a moving line, see Lamb, 
Calculus, Arts. 99, 101 ; Gibson, Calculus, §§ 128, 129 ; Echols, Calculus, 
Arts. 163, 164. 

209. Lengths of curves: rectangular coordinates. Let it be re- 
quired to find the length of an arc 
PQ of the curve whose equation is 
y =f(x), or F(x, y) = 0. Let P, Q 
be the points (x 1} y x ), (x 2 , y 2 ) respec- 
tively, and denote the length of PQ 
by s. 

Suppose that chords like VW 
are inscribed in the arc from P to 
Q. Through V draw VN parallel 
to the »-axis, and through W draw ° 



Q(xo,y 2 ) 




P(Zl,1/l) 



Fig. 12i>. 



208, 209.] LENGTHS OF CURVES. 371 

TTJV parallel to the y-axis. Let V be (a;, y) and TP be (cc + Ax, 
y + Ay). Then F-2V==Ajc, TTJV=Ay, and 



chord FTF= V (Az) 2 + (A?// (1) 



Now suppose that Ax, and consequently Ay, approach zero; 
then the arc VW and the chord VW both become infinitesimal. 
The smaller the chords VW from P to Q are taken, the more 
nearly will their sum approach to the length of the arc PQ. The 
difference between their sum and the length of PQ can be made 
as small as one pleases, simply by decreasing the arcs. Thus : 

s = limit of sum of chords VW when these chords become 
infinitesimal * 



4MW 



Ax 



= j"** V 1 + (H) 2 ' dx * (Definitions, Arts - 22 > 23 > 166 ( 4 ) 
Similarly, from form (3), 



H::Mm*-*v- ® 



Note 1. The quantities under the integration sign in (4) and (5) are the 
infinitesimal elements of length in rectangular coordinates. The differential 
of the arc also has the same forms (Art. 67 c) ; see Note 1, Art. 208. 

Note 2. In (4) the integrand must be expressed in terms of x ; in (5) in 
terms of y. 

Note 3. The process of finding the length of a curve is often called the 
rectification of the curve ; for it is equivalent to getting a straight line of the 
same length as the curve, f 

* For rigorous proof of this, depending on elementary algebra and geom- 
etry, see Rouche" et Comberousse. Traite de Geometrie (1891), Part I., § 291. 
For a proof of the same principle and for interesting remarks on the length 
and rectification of a curve, see Echols, Calculus, Arts. 165, 172. 

t The semi-cubical parabola was the first curve that was ever rectified 
absolutely. William Neil (1637-1670), a pupil of Wallis at Oxford, found 
the length of any arc of this curve in 1657. This was also accomplished 



372 INTEGRAL CALCULUS. [Ch. XXV. 

Note 4. It can be shown : (a) that the difference between an infinitesimal 
arc and its chord is an infinitesimal of an order at least three lower ; (&) that 
the limit of the sum of an infinite number of infinitesimal arcs is the same 
as the limit of the sum of the chords of these arcs. (See Infinitesimal Cal- 
culus, Art. 19, Ex. 3, Note, and Art. 21, Theorems A and B.) 

EXAMPLES. 

2 2 2 

1. Find the length of the four-cusped hypocycloid x 3 + y 3 = a ¥ . 



Length of a quadrant = f * \|l + I^Ydx. (1) 

On differentiation, - af* + - y~^ = ; whence ^ - 



3 3 dx dx \x 



'.-. a quadrant = f ° \ 1 +^dx= C a yj x3 + V* dx = f °^ 

t/0 j3 «/o x 3 «-^° X 3 



3 



dx = - a. 

X 3 «^° X 3 «^° X 3 

.*. length of hypocycloid = 4 x|fl=6a.- 

Note 5. The hypocycloid, sometimes called the astroid, may also be 
represented by the equations x = a cos 3 0, y = a sin 3 0. (This may be veri- 
fied by substitution.) On using these equations it follows that 

dx = — 3 a cos 2 sin d0, dy = 3 a sin 2 cos d0, 

dy 
whence -?- — — tan 0. 

dx 

Thence (1) becomes : 

/* Q—Q . 

length of quadrant = — \ n Vl + tan 2 • 3 a cos 2 sin dd 



=*.£ 



sin cos dd = — , as before. 

2 



(Ex. Show that the area of the hypocycloid x = a cos 3 0, y = a sin 3 6 
is f 7ra 2 ; and that the volume generated by its revolution about the x-axis is 
T % 2 5 7ra 3 , as obtained otherwise in Art. 182, Ex. 20.) 

2. Find the lengths of the following : 

(1) The circle x' 2 + y 2 == a 2 . (2) The arc of the parabola y' 2 = 4 ax, (a) from 
the vertex to the point (asi, y{) ; (6) from the vertex to the end of the latus 

independently by Heinrich van Heuraet in Holland. The second curve to 
be rectified was the cycloid. This was effected by the famous architect, 
Sir Christopher Wren (1632-1723), in 1673, and also by the French mathe- 
matician, Pierre de Fermat (1601-1665). 



209, 210.] 



LENGTHS OF CURVES. 



373 



rectum. (3) (a) The arc of the cycloid x = a (6 — sin 0), y = a (I — cos 0) 
from 6 = 6q to 6 = 6>i ; (b) a complete arch of this cycloid. (4) The arc of 



the catenary 



(e a + e a ), («) from the vertex to (xi, yi); (b) from the 



vertex to the point for which x = a. 

3. Eind the whole length of the curve 
deduce the length of the hypocycloid. 



= 1. Thence 



4. Show that in the ellipse x = a sin <f>, y = b cos <f>, being the com- 
plement of the eccentric angle, the arc s measured from the extremity of the 

minor axis is a \ Vl — e 2 sin 2 d<j>, e being the eccentricity. (This integral is 
called "the elliptic integral of the second kind.") Then show that the perim- 
eter of an ellipse of small eccentricity e is approximately 2 to, 



(-!) 



210. Lengths of curves : polar coordinates. Let it be required to 

G find the length of an 

Q(r tf ,0 2 ) arc PQ of the curve 
f(r, 6) = 0. Let P and 
Q be the points (i\, X ), 
(r 2 , 6 2 ), respectively, and 
denote the length of arc 
PQ by s. Suppose that 
chords like VW are in- 
scribed in the arc from 
P to Q. Let V and W 
be denoted as the points 
(r, 6), (r + Ar, 6 + &0), 
respectively. Then, from Eq. (2) Art. 67 d, 

\( sinA0\ 2 




chord VW= 



\[r 



A6 



sin^-A# • -i a a , A A 2 > n /iN 
-■■■■ ( r \. n . sin \ A0 H • A0. (1) 



1A6» 



AO 



The length of the arc PQ (see Art. 209) is the limit of the sum 
of the lengths of the chords FIT from Pto Q, when these chords 
become infinitesimal, that is when A0 approaches zero. Hence, 
from (1) and the definitions of a derivative and an integral, 



Cj r * 



dry 
dQ) 



(2) 



374 INTEGRAL CALCULUS. [Ch. XXV. 

It can also be shown [see the derivation of resnlt (6), Art. 67 d], 

that . = £^*(£J;)? + l.«Ir. (3) 

Note 1. The quantities under the integration sign in (2) and (3) are the 
infinitesimal elements of length in polar coordinates. The differential of the 
arc also has the same forms, Art. 67 d ; see Note 1, Art. 209. 

Note 2. In (2) the integrand must be expressed in terms of ; in (3), 
in terms of r. 

Note 3. The intrinsic equation of a curve. See Appendix, Note B. 

EXAMPLES. 

1. Find the length of the cardioid r = a(l — cos 0). 



The substitution of the value of r and — in the integrand and simplifica- 

dd 
tion, give 

s = 2 a V2 ( v Vl - cos 6 dd = 4 a f " sin - dd = 8 a. 

Jo Jo 2 

2. Find the lengths of the following : 

(1) The circle r = a. (2) The circle r = 2asin0. (3) The curve 

a 

r = asin 3 -- (4) The arc of the equiangular spiral r = ae ecota , (a) from 
o 

6 = to 6 = 2 ?r ; (6) from 6 = 2 ?r to 6 = 4 tt. (5) The arc of the spiral of 

Archimedes r = ad from (n, #i) to (r 2 , 6 , 2 ). (6) The arc of the parabola 

r = a sec 2 -, (a) from = to 6 = On (b) from = --to0= + -> 

2 v ' K 2 2 



211. Areas of surfaces of revolution. 

Note 1. Geometrical Theorem. Let KL and BS (Fig. 131 a) be in the 
same plane. In elementary solid geometry it is shown that if a finite straight 
line KL makes a complete revolution about BS, the surface thu^ generated by 
KL is equal to 2 w TM • KL, in which TM is the length of tht, perpendicular 
let fall on BS from T 7 , the middle point of KL. 

Suppose that an arc PQ of a curve y =f(x) revolves about the 
a;-axis, and that the area of the surface thus generated is required. 



210, 211.] 



AREAS OF SURFACES. 



375 



Let P and Q be the points (x^ ?/i) and (x 2 , y 2 ) respectively. Sup- 
pose that PQ is divided into small arcs such as KL, and denote 
K and L as the points (x, y) and (x + Ax, y -f- Ay) respectively. 





Q(z 2 ,?/ 2 ) 



R 



M 
Fig. 131a. 



SO 



M 
Fig. 131 6. 



Draw the chord KL, and from T, the middle point of this chord, 
draw TM at right angles to the a>axis. Then the area generated 
by the chord KL when the arc PQ revolves about the x-axis 

= 2ttTM.KL 



2 tt 0/ + i Ay)yjl + f^j • Ax, (Note 1.) 

The smaller the chords KL are taken, the more nearly will the 
surfaces generated by them approach coincidence with the surface 
generated by the arc PQ, and the difference between area of the 
latter surface and the sum of the areas of the former surfaces 
can be made as small as one pleases by decreasing Ax. Accord- 
ingly, the area of the surface generated by the arc PQ is the 
limiting value of the sum of the areas of the surfaces generated 
by the chords KL (from P to Q) when these chords become 
infinitesimal. That is, area of surface generated by JPQ 



= Hm A ^5 2 * (y ■ + i A y) V 1 + {% 



Ax 



(Definitions of derivative 



a f* 2 /-. , fdy\2^ (Definitions of d 
-H^'V+U)*'- and integral.) 



(1) 



(2) 



376 



INTEGRAL CALCULUS. 



TCh. XXV. 



If the length of the chord KL be denoted by -Jl + (— Ya?/ 
this integral takes the form ™ K^vJ 



surface 



-i r fiJi+(&\% 



dy 



(3) 



Note 2. Each of the expressions to be integrated in (2) and (3) may be 
denoted by 2-irijds [Art. 67 /(9)], and is called an element of the surface 
of revolution. 

If PQ is revolved about the y-axis, the element of surface is 2 trx ds ; 
and the surface 

= 2 . I""* 1 "* v ds = 2 I C>=°> X S^f d y 



x=x 1 ,ij=tj l 



dy 



\GW 



i + i d -nYdx. 

dxj 



(4) 



The questions, whether to use form (2) or (3), and which of (4) to employ, 
are decided by convenience and ease of working. (See Art. 208, Note 1, and 
Art. 67/.) 

Note 3. In a similar manner it can be shown that the area of the surface 
generated by the revolution of an arc of a curve about any straight line in 
the plane of the arc, is ~ 

2 7T ( Ids, (5) 

in which ds denotes an infinitesimal arc of the curve, I the distance of this 
infinitesimal arc from the straight line, and e x and e 2 are coordinates of some 
kind that denote the ends of the revolving arc. An illustration is given in 
Ex. 4. 

EXAMPLES. 
1. Find the surface generated by the revolution of the hypocycloid 



SB* 


Surface 


about the x-axis. 




Jjc=0 

= 4 7rf X 
Jx= 


2ir.PN.ds 




i 



ra 2 2 3 n 3 

= 4tT j («3_ £3)2 . O^dX 



X 3 



(See Art. 209, Ex. 1.) 

1 fa 2 2 3 2 2 

= -6rra*\ («3_ x*)?d(a* -X s ) 




3. X 



Fig. 132. 



211.] 



AREAS OF SUBFACES. 



377 



In this case au easier integral is obtained by expressing the surface in 
terms of y and cZy, as in form (3) . Thus, 



Surface = 2 - 2ir \ y-W 1 + ( 



dx\ 2 



dy 



■a^\%j 3 dy 



2. Calculate the surface of the hypocycloid in Ex. 1, using the equations 
x = a cos 3 d, y = a sin 3 9. 

3. Derive formula (5). 

4. The cardioid r = «(1 — cos0) revolves about the initial line : find the 
area of the surface generated. 



Surface 



Je=o 



PN- 



Now PJY—r sin 6 = a(l — cos0)sin 0, and ds = av^vl — cosddd (see 
Ex. 1, Art. 210). 



0=T 




0-0 



Fig. 133. 



surface = 2V2ira 2 (" (1 - cos 0)~* sin Odd = f_^l7ra 2 (l - cos0)?T 



5. Find the area of the spherical surface generated by the revolution of a 
circle of radius a about a diameter. 

6. A quadrant of a circle of radius a revolves about the tangent at one 
extremity. What is the area of the curved surface generated ? 

7. Calculate the area of the surface of the prolate spheroid generated by 
the revolution of the ellipse b 2 x 2 + a 2 y 2 — a?b 2 about the x-axis. 

8. In the case of an arch of the cycloid x=a(0— sin0), y=a(l — cos0), 
compute : (1) the area between the cycloid and the x-axis ; (2) the volume 
and the surface generated by its revolution about the x-axis ; (3) the volume 
and the surface generated by its revolution about the tangent at the vertex. 

9. Find the volume and the surface generated by revolving the circle 
* 2 f (y — &) 2 = «" 2 5 (ft > a), about the x-axis. 



378 INTEGRAL CALCULUS. [Ch. XXV. 

10. Find the area of the surface generated by the revolution of the arc 
of the catenary in Ex. 6, Art. 182. 

11. The arc of the curve r = asin2 0, from 6 = to =- Oi.e. the 

4 
first half of the loop in the first quadrant), revolves about the initial line : 
find the area of the surface generated. What is the area of the surface 
generated by the revolution of the second half of the same loop about the 
same line ? 

12. A circle is circumscribed about a square whose side is a. The smaller 
segment between the circle and one side of the square is revolved about 
the opposite side of the square. Find the volume and the surface of the 
solid ring thus generated. 

212. Areas of surfaces whose equations have the form z =f(oc 9 y) 

or F(x 9 y 9 z) = 0. It is shown in solid geometry that: 

(a) The cosine of the angle between the xy-plane and the tangent plane 
at any point (x, y, z) on such a surface, supposed to be continuous, is 

{ 1+ (I) 2+ (I)T- . « 

(b) The area of the projection of a segment of a plane upon a second 
plane is obtained by multiplying the area of the segment by the cosine of 
the angle between the planes. 

It follows from (a) and (6) that : 

(c) If there be an area on the xy-plane equal to A, then A is the area 
that would be projected on the xy-plane by an area on the tangent plane at 
(x, y, z) which is equal to 

^+(gr+(ir- 

(See C. Smith, Solid Geometry, Arts. 206, 26, 31 ; Murray, Integral Calcu- 
lus, Art. 75.) 

Let z —fix, y) be the equation of a surface BFCRAGB [Fig. 123] whose 
area is required. Let P(x, y, z) be any point on this surface, and Pi the 
point (x, y, 0) vertically below P. Let PiQi be a rectangle in the x?/-plane 
having its sides equal to Ax and Ay respectively, and parallel to the x- and 
?/-axes. Through the sides of this rectangle pass planes perpendicular to the 
xy-plane, and let these planes make with the surface the section PQ, and 
with the tangent plane at P the section PQ 2 . (QiQ produced is supposed 
to meet in Q 2 the tangent plane at P.) 

Then, area P\Q\ = Ax • Ay. 



**-Vi*(g)V(gr 



Hence, by (2), area PQ 2 = V 1 + ^ + ?) ' A V ' Ax 



212.] AREAS OF SURFACES, 379 

Now the smaller Ax and Ay become, the more nearly will the section PQ 2 
on the tangent plane at P coincide with the section PQ on the surface. 
Accordingly, the more nearly will the sum of the areas of sections like PQ 2 
on the tangent planes at points taken close together on the surface, become 
equal to the area of the surface ; moreover, the difference between this sum 
and the area of the surface can be made as small as one pleases. Con- 
sequently, the area of the surface is the limit of the sum of the areas of 
these sections on the tangent planes when these sections become infinitesimal. 

That is, 

area BFCX^B J£» g ^l + (g) ' + (g)' • « *. 

Note. The integral [ P =S(? a/i 4- f — V + (^Ydy~]dx gives the area 



-UrsMSHS)'*] 



of the strip or zone RGL, and the integral \ BGLdx gives the sum of 
these zones from BOC to A. 

EXAMPLES. 

1. Find the area of the portion of the surface of the sphere in Ex. 7, 
Art. 204, that is intercepted by the cylinder. 

The area required = 4 area A VBLA (Fig. 125) . In this figure, the equation 
of the sphere is x 2 + V 1 + z 2 = a 2 , 

and the equation of the cylinder is x 2 + y 2 = ax. 

The area of a strip L F, two of whose sides are parallel to the si/-plane, will 
first be found ; then the sum of all such strips in the spherical surface 
AVBLA will be determined. 

— — ir£rNgy+(f) 2 ]^ 

Since the required surface is on the sphere, the partial derivatives must be 
derived from the equation of the sphere. 

dx z dy z' 



dx. 



Accordingly, 



dx) \dy) z 2 z 2 z 2 a 2 -x 2 -y* 



Also, BK = Vax — x 

Wax-x* 



J 'a rVax-x2 n 
\ a dy dx 

JO i//,2 _ rri _ o,2 



a I sin- 1 — y 

JO L >//y2 _ ^.2 Jo 



dx 



= a \ sin -1 \ — - — dx. 
Jo \ a + x 



380 INTEGRAL CALCULUS. [Ch. XXV. 

This integral can be evaluated by integrating by parts. The integration 

can be simplified by means of the substitution sin z =\l- — ^— It will be 

X a + x 
found that area required = 4 area AVE LA = 2 (ir - 2) a 2 = 2.2832 a 2 . 

2. Find the area of the surface of the cylinder intercepted by the sphere 
in Ex. 7, Art. 204. 

3. By the method of this article, find the surface of the sphere x 2 -f y 2 

+ z 2 = a 2 . 

4. A square hole is cut through a sphere of radius a, the axis of the 
hole coinciding with a diameter of the sphere : find the volume removed and 
the area of the surface cut out, the side of a cross-section of the hole being 2 6. 

5. Find the area of that portion of the surface of the sphere inter- 
cepted by the cylinder in Ex. 4, Art. 205. 

213. Mean values. In Art. 168 it has been stated that if the 
curve y=f(x) be drawn (Fig. 101), and if OA = a and OB = b, 
then, of all the ordinates from A to B, 

I f(x) dx 



,, i area APQB 

the mean value = ■ -^— = ' 



AB b 



(1) 



Result (1) can be derived in the following w r ay wdiich has 
also the advantage of being adapted for leading up to a more 
general notion of mean value. The mean value of a set of quan- 
tities is defined as 

the sum of the values of the quantities 
the number of the quantities 

For instance, if a variable quantity takes the values 2, 5, 7, 9, 

the mean of these values is -^ or 5f . 

4 

Now take any variable, say x, and suppose that f(x) is a con- 
tinuous function, and let the interval from x = a to x = b be 
divided into n parts each equal to Ax, so that n Ax = b — a. Let 
the mean of the values of the function for the n successive values 

a, a + Ax, a + 2 Ax, •••, a + n — lAx, 

be required. The corresponding n successive values of the func- 
tion are /( ^ f ^ ^ f ^ + g ^ ^ f ^ + — j _ A ^ 



213.] MEAN VALUES. 

Hence, mean value of function 



_ f(a) +/(« + Aa?) ±f(a + 2 Aa;) + ; ; ; +/(a + n-l- Aa;) 



381 



(2) 



Now ?i Aa; = b — a, whence w = 
mean value 



Aa; 



Substitution in (2) gives 



_ /(q)A.r+/(a4- Aa;)Aa?+/(a+2 AaQAa; j \-f(a+n— 1 AaQAa; 

Finally, let the mean of all the values that f(x) takes as x varies 
from a to b be required. In this case n becomes infinitely great 
and Aa; becomes infinitesimal; accordingly [Art. 166 (2), (3)] 



(3) becomes 



mean value 



f V(as) doc 

__ Ja 

b — a 



(4) 



as already represented geometrically in Art. 168. 



Note 1. Keference for collateral reading. Echols, Calculus, Arts. 
150-152. 

EXAMPLES. 

1. Find the mean length of the ordinates of a semicircle (radius a). 
the ordinates being erected at equidistant intervals on the diameter. 

Choose the axes as in Fig. 134. Then the equation of the circle is 
x 2 + y 2 = a 2 . Let PN denote any of the ordi- 
nates drawn as directed. 




Mean value 



£ 



PN-dx 



a — (— a) 



j" 

J —a 



Va 2 — x 2 dx 



2a 



ira< 



2.2a 

2. Find the mean length of the ordinates of a 
semicircle (radius a), the ordinates being drawn at 
equidistant intervals on the arc. 

Let PN be any of the ordinates drawn at equi- 
distant intervals on the arc, that is, at equal incre- 
ments of the angle 6. 

•0=77 



= .7854 a. 




Mean value = 



V 



PN-dd 



£« 



sin e dd 



2a 



382 INTEGRAL CALCULUS. [Ch. XXV. 

Note 2. A slight inspection will show that it is reasonable to expect the 
results in Exs. 1, 2, to differ from each other. 

Suggestion : Draw a number of ordinates, say 4 or 6 or 8, as specified 
in Ex. 1, and compare them with the ordinates of equal number drawn as 
specified in Ex. 2. 

3. Find the average value of the following functions: (1) 7cc 2 +4x — 3 
as x varies continuously from 2 to 6 ; (2) x % — 3 x 2 + 4 x + 11 as x varies from 
— 2 to 3. Draw graphs of these functions. 

4. Find the average length of the ordinates to the parabola y 2 = 8 x 
erected at equidistant intervals from the vertex to the line x = 6. 

5. (1) In Fig. 108 find the mean length of the ordinates drawn from 
O-ZVto the arc OML, and the mean length of the ordinates drawn from ON to 
the arc ORL. (2) In Fig. 107 find the mean length of the abscissas drawn 
from OY, (a) to the arc OR; (6) to the arc RL ; (c) to the arc ORL. 
(3) In Fig. 109 find the mean ordinate from OL, (a) to the arc TKN '; (b) to 
the arc TGM. 

6. (1) In the ellipse whose semiaxes are 6 and 10, chords parallel to 
the minor axis are drawn at equidistant intervals : find their mean length. 
(2) In the ellipse in (1) find the mean length of the equidistant chords that 
are parallel to the major axis. (3) Do as in (1) and (2) for the general case 
in which the major and minor axes are respectively 2 a and 2 b. 

7. On the ellipse in Ex. 6, (3), successive points are taken whose eccen- 
tric angles differ by equal amounts: find the mean length of the perpen- 
diculars from these points, (1) to the major axis ; (2) to the minor axis. 

8. In the case of a body falling vertically from rest, show that (1) the 
mean of the velocities at the ends of successive equal intervals of time, is one- 
half the final velocity ; (2) the mean of the velocities at the ends of succes- 
sive intervals of space, is two-thirds the final velocity. (The velocity at the 
end of t seconds is gt feet per second ; the velocity after falling a distance 
s feet is V2 gs feet per second.) 

9. A number n is divided at random into two parts : find the mean value 
of their product. 

10. Find the mean distance of the points on a circle of radius a from 
a fixed point on the circle. 

The interval b — am (1) and (4) through which the variable x 
passes is called the range of the variable, and dx is an infini- 
tesimal element of the range. In (1) and Ex. 1 the range is a 
particular interval on the #-axis. In Ex. 2 the range is a certain 
angle, namely ?r; in Ex. 8 (2) the range is a vertical distance j in 



213.] MEAN VALUES. 383 

Ex. 8 (1) the range is an interval of time. There are various 
other ranges at (or for) whose component parts a function may 
take different values. For instance, a curved line as in Ex. 10, a 
plane area as in Exs. 11, 13 ; a curved surface as in Ex. 15 (1) ; a 
solid as in Exs. 16, 17. The definition of mean value [or result 
(4)] may be extended to include such cases, thus : 

lim ^ {(value of function at each infini- 
tesimal element of the range) x (this 

the mean value of a func- ) _ infinitesimal element)} # 

tion oyer a certain range J the range 

11. Find the mean square of the distance of a point within a square 

(side = a) from a corner of the square. 

In this case "the range" extends over a square. 
Choose the axes as shown in Fig. 136. Take any point 
P (ac, y) in the range, and let its distance from be 
d. At T let an infinitesimal element of the range 
be taken, viz. an element in the shape of a rectangle 
whose area is dy dx. Now d 2 = x 2 -f y 2 . .*. mean 
value of d 2 for all points in 
Fig. 136. 

( a (\x 2 + y 2 )dydx 

OACB = ^ZQ _ 2 a 2 # 

area of square 

12. Find (1) the mean distance, and (2) the mean square of the distance, 
of a fixed point on the circumference of a circle of radius a from all points 
within the circle. (Suggestion : use polar coordinates.) 

13. Find (1) the mean distance, and (2) the mean square of the distance, 
of all the points within a circle of radius a from the centre. 

14. Find the mean latitude of all places north of the equator. 

15. For a closed hemispherical shell of radius a calculate (1) the mean 
distance of the points on the curved surface from the plane surface ; (2) the 
mean distance of the points on the plane surface from the curved surface, 
distances being measured along lines perpendicular to the plane surface. 

16. Calculate (1) the mean distance, and (2) the mean square of the dis- 
tance, of all points within a sphere of radius a, from a fixed point on the 
surface. 

17. Calculate (1) the mean distance, and (2) the mean square of the dis- 
tance, of all points within a sphere of radius a, from the centre. 



1 

B 








C 


" 




m 








L 


*y 







A 


D 











A X 




u 





^ 



381 



INTEGRAL CALCULUS. 



[Ch. XXV 



18. Find (1) the mean distance, and (2) the mean square of the distance, of 
all points on the surface of a sphere of radius a, from a fixed point on the surface, 

19. Find (1) the mean distance, and (2) the mean square of the distance, of 
all points on a semi-undulation of the sine curve y = a sin x, from the x-axis. 

214. Note to Art. 104. Proof of (6). Let iTbe the given curve 

y = f(x), and E its evolute. 

Let C x be the centre of curva- 
ture for A Y , and C 2 for A 2 . Denote 
any point in iTby (x, y), the radius 
A^x^Vz) of curvature there by R, the cor- 
responding centre of curvature in 
E by (a, ft), the points A 1} A 2 , C 1} 
C 2 , by fa 2/x), (x 2 , y 2 ), («,, ft), (a 2 , 
ft), respectively, the radii of cur- 
vature A l C l and A 2 C 2 by R 1 and R 2 . 
It will now be shown that 
length of arc C1C2 = JB2 _ Mu 




Fig. 137. 



Arc 0,0,= C P *Jl +(— Y • dft (See Art. 209.) (1) 

On substituting the value of — from (3) Art. 104, and the 

value of dp derived from (1) Art. 104, and noting that 

x — x Y when ft = ft, and x = x 2 when ft = ft, 

Equation (1) becomes 



arcc ^=li:N i+ (l) ! 



(7jc V dor 



1 + 



cU- 3 



^X2 

cfar\ 



cto. (2) 



dR 



Differentiation of R in Art. 101, Eq. (1), will snow that — - is 

dx 

the same as the integrand in (2). Then, since R = R X when 

x = a? 1? and R = R 9 when x = x 2 , and — - dx = dR (Art. 27), Equa- 

dx 

tion (2) becomes 

arc CiC 2 = C =X2 ^dx= C =H dR= C =R " dR = R 2 -R x . 

J %=x x dx J x=% x Jr=r x 

N,B. On lengths of curves in space see Appendix, Note C. 



CHAPTER XXVI. 

NOTE ON CENTRE OF MASS AND MOMENT OF 
INERTIA. 

N.B. For a full explanation and discussion of the mechanical terms in 
this note, see text-books on Mechanics. 

215. Mass, density, centre of mass. For this note the following 
definition of mass may serve : The mass of a body is the quantity 
of matter which the body contains* The principal standards of 
mass are two particular platinum bars; the one is the "imperial 
standard pound avoirdupois," which is kept in London, and the 
other is the " kilogramme des archives," which is kept in Paris. 

Note. The weight of a body is the measure of the earth's attraction upon 
the body, and depends both on the mass of the body and its distance from 
the centre of the earth. The same body, while its mass remains constant, 
has different weights according to the different positions it takes with respect 
to the centre of the earth. 

The density of a body is the ratio of the measure of its mass to the measure 
of its volume ; that is, the density is the number of units of mass in a unit of 
volume. The density at a point is the limiting value of the ratio of (the 
measure of) the mass of an infinitesimal volume about the point to (the 
measure of) the infinitesimal volume. A body is said to be homogeneous when 
the density is the same at all points. If a body is not homogeneous, the "den- 
sity of a body," defined above, is the average or mean density of the body. 

Centre of mass. Suppose there are particles whose masses are m 1? 
m 25 m 3> ■"> an( i whose distances from any plane are, respectively, 
d lf d 2 , d 3 , •••. Let a number D be calculated such that 

D = mA + m^ + fflg^- . ie let D = ^md 
mi + ^2 -h wi 3 + • • • %m 

For any given plane, D evidently has a definite value. 

* A real definition of mass, one that is strictly logical and fully satisfac- 
tory, is explained in good text-books on dynamics and mechanics. (For 
example, see MacGregor, Kinematics and Mechanics, 2d ed., Art. 289.) 

385 



386 INTEGRAL CALCULUS. [Ch. XXVI. 

If («„ y 1} z x ), (x 2 , y 2 , z 2 ), (x 3 , y s , %),•••, respectively, be the coordi- 
nates of these particles with respect to three coordinate planes at 
right angles to one another, then the point (x, y, z), such that 

-_2mx -_-%my -_%mz m 

*""*P y ~^> *~~W (1) 

is called the centre of mass of the set of particles. 

If the matter " be distributed continuously " (as along a line, 

straight or curved, or over a surface, or throughout a volume), and 

if Am denote the element of mass about any point (x, y, z), then, 

on taking all the points into consideration, equations (1) may be 

written : 

x = H^^-Am and s i m ii ar i y f or y and z. (2) 

linwoSAm' J y w 

From (2), by the definition of an integral, 

( ac dm \ y dm \ z dm 

* = } - c , V = } c ,1 = 4 (3) 

\ dm \ dm \ dm 

If p denote density of an infinitesimal dv about a point, then 

dm = pdv (4) ; and, on integration, m = J p dv. (5) 

Ex. Write formulas (3), putting p dv for dm. 

Suppose that the body is not homogeneous; that is, suppose 
that the density of the body varies from point to point. Let p 
denote the density at any point (x, y, z), let dv denote an infini- 
tesimal volume about that point, and let p denote the average or 
mean density of the body. Then 

mass of body J p ^ v 
9 



vol. of body C dv 



Note. The term centre of mass is used also in cases in which matter is 
supposed to be concentrated along a line or curve, or on a surface. In these 
cases the terms line-density and surface-density are used. 



215.] 



CENTRE OF MASS. 



387 



EXAMPLES. 

1. In a quadrant of a thin elliptical plate whose semi-axes are a and &, 
the density varies from point to point as the product of the distances of each 

point from the axes. Find the mass, 
the mean density, and the position of 
the centre of mass, of the quadrant. 
Choose rectangular axes as in the figure. 
At any point P(x, y), let p denote the 
density and dm denote the mass of a 
rectangular bit of the plate, say, dx • dy. 
Let M denote the mass, p the mean 
density, and (x, y) the centre of mass, 
of the quadrant. 

Now dm = p dx dy. But pccxy ; i.e. 



1 

L 
1) 


X 


Qdy \ 






dx \ 
V \ 





« a ^A 



Fig. 138 
p = kxy, in which k denotes some constant. 

mass of quadrant _ \k a 2 b' 2 
volume of quadrant \ nab 



dm = \ \ a kxy dy dx — \k a 2 b 2 . 

Jx=0 ^i/=0 



Also, 



kab 
2tt 



Here 



Similarly, 



, I 



p • x • dv k 



f-f; 

Jo Jo 



Ya2-x2 



x 2 y dy dx 



_^kam_ 8 



ka 2 b 2 



\ p • dv M 

y = ^ 6. Hence, centre of mass is ( T 8 5 a, r 8 3 &). 



2. Find the centre of mass of a solid 
hemisphere, radius a, in which the density 
varies as the distance from the diametral 
plane. Also find the mean density. 

Symmetry shows that the centre of mass 
is in OY. 

Take a section parallel to the diametral 
plane and at a distance y from it. 

The area of this section 

= it • CP 2 = 7r(a 2 - y 2 ). 

For this section, p oc y, i.e. p — ky, say. 




Fig. 139. 



Then 



JjV ■ y • ir{a 2 - y 2 )dy kir^y 2 {a 2 - y 2 )dy 
§*pir(a 2 -y 2 )dy k-w^y{a 2 



a. 



y 2 )dy 



M 



vol. | ira 3 



Also p = — r _ -^ -— j 

This is the density at a distance | a from the diametral plane, 



388 



INTEGRAL CALCUL US. 



[Ch. XXVI. 



3. The quadrant of a circle of radius 
at one extremity. Find the position 
thus generated. In this case let the 
"surface-density" be denoted by p. 
Symmetry shows that the mass-centre 
is in the line PL. Let y denote the 
distance of the mass-centre from OX 

In PL take any point N, at a dis- 
tance y, say, from OX. Through N 
pass a plane at right angles to PL, 
and pass another parallel plane at an 
infinitesimal distance dy from the first 
plane. These planes intercept an infini- 
tesimal zone, of breadth ds say, on the 
surface generated. 

Area of this zone = 2 tt • ON- ds = 2 tt(MN— MC)ds. 



a revolves about the tangent 
of the mass-centre of the surface 

Y 




Now, at C (x, y) 



x 2 + 



Accordin 



ingly, ds = ^l + (^Y-dy = 



dy 



Va 2 - y 2 



Hence, area zone = 2 ir (a — Va 2 - y 2 ) a dy = 2wa1 a — 1 ] dy. 

Va' 2 - y 2 V Va 2 - y 2 I 



C V a py- (2tt.CN -ds) 2 
Jy=o 



* ap $o : 



Va 2 - v 2 



~l)dy 



p • area zone 2 w ap \ [ 1 1 1 

J» Wa^V 2 J 



= .876 a. 



4. In the following lines, curves, surfaces, and solids, find the posi- 
tion of the centre of mass ; and, in cases in which the matter is not dis- 
tributed homogeneously, also find the total mass and the mean density 
("line-density," "surface-density," or "density," as the case may be). 
(The density is uniform, unless otherwise specified.) 

(1) A straight line of length I in which the line-density varies as (is k 
times), (a) the distance from one end ; (6) the square of this distance ; (c) the 
square root of this distance. 

(2) An arc of a circle, radius r, subtending an angle 2 a at the centre. 

(3) A fine uniform wire forming three sides of a square of side a. 

(4) A quadrantal arc of the four-cusped hypocycloid. 

(5) A plane quadrant of an ellipse, semi-axes a and 6. 



215.] CENTRE OF MASS. 389 

(6) The area bounded by a semicircle of radius r and its diameter, 
(a) when the surface density is uniform ; (6) when the surface density at 
any point varies as (is k times) its distance from the diameter. 

(7) The area bounded by the parabola Vx + Vy = Va and the axes. 

(8) The cardioid r = 2 a (1 - cos 0). 

(9) A circular sector having radius r and angle 2 a. 

(10) The segment bounded by the arc of the sector in Ex. (9) and its chord. 

(11) The crescent or lune bounded by two circles which touch each other 
internally, their diameters being d and §#, respectively. 

(12) The curved surface of a right circular cone of height h, (a) when 
the surface density at a point varies as its distance from a plane which passes 
through the vertex and is at right angles to the axis of the cone ; (b) when 
the surface density is uniform. 

(13) A thin hemispherical shell of radius a, in which the surface density 
varies as the distance from the plane of the rim. 

(14) A right circular cone of height h in which, (a) the density of each 
infinitely thin cross-section varies as its distance from the vertex ; (6) the 
density is uniform. 

(15) Show that the mass-centre of a solid paraboloid generated by revolving 
a parabola about its axis, is on the axis of revolution at a point two-thirds the 
distance of the base from the vertex. 

(16) A solid hemisphere of radius r, (a) when the density is uniform ; 
(&) when the density varies as the distance from the centre. 

(17) Show that the mass-centre of the solid generated by the revolution 
of the cardioid in Ex. (8) about its axis, is on this axis at a distance § a from 
the cusp. 

(18) If the density p at a distance r from the centre of the earth is given 

by the formula p = p -, in which p and k are constants, show that the 

kr 
earth's mean density is . , n 7 „ 7 „ 

J o sin k B — kB cos kB 

o Po 1 

&B* 

in which B denotes the earth's radius. (Lamb's Calculus.) 

[Answers : (1) f I from that end, M = \ kl 2 , p = ^kl; (6) f I, M = % kP, 

p = \kl 2 ; (c) f Z, M = | kl?, p =%kl2. (2) On radius bisecting the arc at dis- 
tance r from centre. (3) At a distance i a from the centre of the 

square. (4) Point distant f a from each axis. (5) Point distant — from 

axis 2 a, — ' from axis 2 b. (6) (a) On middle radius, at point distant — 
3 7r 3ir 

from the diameter ; (6) On middle radius, at point .589 a from the diameter, 
mean density = .4244 maximum density. (7) Point distant \ a from each 
axis. (8) The point (?r, f a). (9) On middle radius of sector, at distance 

f r from the centre. (10) On the bisector of the chord, at distance 



390 INTEGRAL CALCULUS. [Ch. XXVI. 

I r : from the centre. (11) On the diameter through the point 

3 a - sin a cos v J b * 

of contact and distant if d from that point. (12) (a) On the axis, at distance 

| h from the vertex ; (6) on axis, at distance f h from vertex. (13) On the 

radius perpendicular to the plane of the rim, at a distance f a from the centre. 

(14) (a) On the axis, | h from the vertex ; the mean density is the same as 

the density at the cross-section distant f h from the vertex ; (b) on the axis, 

at a distance f h from the vertex. (16) (a) On a radius perpendicular to the 

base, at a distance .375 r from it; (6) on radius as in (a), at distance Ar 

from the base.] 

216. Moment of inertia. Radius of gyration. These quantities are 
of immense importance in mechanics and its practical applications. 

Moment of inertia. Let there be a set of particles whose masses 
are, respectively, m lf m 2 , m 3 , • • •, and whose distances from a chosen 
fixed line are, respectively, r 1} r 2 , r 3 , •••. The quantity 

m x r? + m 2 r 2 2 + m 3 r 3 2 -\ , i.e. 5 mr% (1) 

is called the moment of inertia of the set of particles with respect 
to the fixed line, or axis, as it is often called. It is evident that 
for any chosen line and system of particles the moment of inertia 
has a definite value. In what follows, the moment of inertia will 
be denoted by I. 

It can be shown, by the same reasoning as in Art. 215, that 
definition (1) can be extended to the case of any continuous dis- 
tribution of matter (whether along a line or curve, or over a sur- 
face, or throughout a solid) and any chosen axis; thus, 



( r2 dm, 



in which r denotes the distance of any point from the axis, and 
dm an infinitesimal element of mass about that point. 

Radius of gyration. In the case of any distribution of matter 
and a fixed line, or axis, the number k, which is such that 



, 2 _ the moment of inertia _ j r dm 
the mass [dm 

is called the (length of the) radius of gyration about that axis. 



216.] 



MOMENT OF INERTIA. 



391 



EXAMPLES. 

1. Find the radius of gyration about its line of symmetry of an isosceles 
triangle of base 2 a and altitude h. 

The density per unit of area will be denoted by p. 




Fig. 141. 

Let P be any point in the triangle, and make the construction shown in 
the figure. Denote NO by y. 



Then k 2 = 2Z PN 2 . p. dxdy over AOC = 2p J^L 



'x=LN 
1x=0 



Sp • dx dy over ABC 



p ah 



Now M= ™ i.e. M=L=JL ; whence LN= <* (h - y). 
AO CO ah h 

.-. &2 = i«!>?: = i a2 . whence k = -^- 
ah 6 V6 

In this example, the moment of inertia is £ a % h. 

2. Show that the moment of inertia of a homogeneous thin circular plate 
about an axis through its centre and perpendicular to its plane is \pir a 4 , in 
which p denotes the surface density, and that its radius of gyration is J aV2. 

On using polar coordinates, I = i r 2 • dm = j r 2 • p • d A — p \ \ V • r dr dd.\ 

Y 3. Find the moment of inertia of a solid 

homogeneous sphere of radius a about a 
diameter, m being the mass per unit of 
volume. Suppose that the sphere is gener- 
ated by the revolution of the semicircle APB 
about the diameter AB. Let rectangular 
axes be chosen as in the figure. At any 
point P(x, y) on the semicircle take a thin 
rectangular strip PN at right angles to AB 




392 INTEGRAL CALCULUS. [Ch. XXVI. 

and having a width Ax. This strip, on the revolution, generates a thin circu- 
lar plate. It follows from Ex. 2, since m is the mass per unit of volume, that 

/of this plate about AB = -ir . PN* . Ax. 

.\ I of sphere = S — tt . fW i Ax from A to B 

2 

= 2 • 5*1 f " («2 _ ^2)2^. _ ^ m7ra 5 # 

2 Jo 
Here, on denoting the mass of the sphere by M, 
M= f mTra 3 ; 
hence, J=fiYa 2 ; 

accordingly, & 2 = § a 2 ; 

and thus, k = .632 a. 

4. Find the moment of inertia and the square of the radius of gyration 
in each of the following cases : 

(Unless otherwise specified, the density in each case is uniform. The 
mass per unit of length, surface, or volume is denoted by m, and the total 
mass by 31.) 

(1) A thin straight rod of length ?, about an axis perpendicular to its 
length : (a) through one end point, (6) through its middle point. 

(2) A fine circular wire of radius a, about a diameter. 

(3) A rectangle whose sides are 2 a. 2 6: (a) about the side 2 6, 
(6) about a line bisecting the rectangle and parallel to the side 2 b. 

(4) A circular disc of radius a : (a) about a diameter, (&) about an 
axis through a point on the circumference, perpendicular to the plane of 
the disc, (c) about a tangent. 

(5) An ellipse whose semi-axes are a and b : (a) about the major axis, 
(b) about the minor axis, (c) about the line through the centre at right 
angles to the plane of the ellipse. 

(6) A semicircular area of radius a, about the diameter, the density 
varying as the distance from the diameter. 

(7) A semicircular area of radius a, about an axis through its centre of 
mass, perpendicular to its plane. 

(8) A rectangular parallelopiped, sides 2 a, 2 6, 2 c, about an edge 2 c. 

(9) A right circular cone (height = 7i, radius of base = 6), about its axis. 

(10) A thin spherical shell of radius a, about a diameter. 

(11) A sphere of radius a, about a tangent line. 

(12) A right circular cylinder (length = I, radius = E) : (a) about its 
axis, (6) about a diameter of one end. 



210] MOMENT OF INERTIA. 393 

(13) A circular arc of radius a and angle 2a: (a) about the middle 
radius, (&) about an axis through the centre of mass, perpendicular to the 
plane of the arc, (c) about an. axis through the middle point of the arc, 
perpendicular to the plane of the arc [Lamb's Calculus, Exs., XXXIX.]. 

[Answers: (1) (a) fmZ 3 , \l 2 ; (6) ^mP, &P. (2) I Ma 2 , \a 2 . 

(3) (a) & 2 = fa 2 ; (b) k 2 = i a 2 . (4) (a) k 2 = \a 2 ; (b) k 2 = f a 2 ; (c) & 2 
= fa*. (5) (a) iitffc 2 ; (6) iJ/a 2 ; (c) J ilf(a 2 -f & 2 ). (6) § itfa 2 , fa 2 . 
(7) fc 2 = U- ^\ a 2 . (8) ^ = f (a 2 + 6 2 ). (9) ^ W7r & *fc, _3_ 6 «. (]0 ) ^ 

= §« 2 . (11) & 2 ='|a 2 . (12) (a) 1=1 MB 2 ; (&) 7= Jf(iJ2 2 + f Z 2 ). 

(13) (a) *» = ia »fl-^^Vi (6) * 2 = a 2 (l-^4^; (c) * 2 = 
/ • \ -. V 2a/ \ a 2 / 

Note. For interesting examples on centres of gravity and moments of 
inertia, see Campbell, Calculus, Chaps. XXXVI., XXXVII, Chandler, Cal- 
culus, Chaps. XXXIII, XXXIV. For discussions on mechanics and exam- 
ples, see Osgood, Calculus, Chap. X., and Campbell, Calculus, Chaps. XXX.- 
XXXV. 



CHAPTER XXVII. 

DIFFERENTIAL EQUATIONS. 

N.B. The references made in this chapter are to Murray, Differential 
Equations. 

217. Definitions. Classifications. Solutions. This chapter is 
concerned with showing how to obtain solutions of a few differen- 
tial equations which the student is likely to meet in elementary 
work in mechanics and physics. 

Differential equations are equations that involve derivatives or 
differentials. Such equations have often appeared in the preced- 
ing part of this book. 

Thus, in Art. 37, Exs. 2, 11, 13, differential equations appear ; Equations 
(1), Art. 63, (2)-(5), Art. 67 (a), (2)-(5), Art. 67 (c), (3)-(6), Art. 67 (d), 
are differential equations ; so also, in Art. 68, are (1) and (2), Ex. 5 ; equa- 
tions in Exs. 13, 14, and some of the equations in Exs. 10, 11 ; several equa- 
tions in Ex. 1, Art. 69 ; Equations (2)-(4), Ex. 1, Art. 73 ; the answers to 
Exs. 2-4, Art. 73; in Ex. 4, Art. 79 ; in Exs. 5-8, Art. 80 ; Equation (8), 
Art. 96 ; etc., etc. 

Differential equations are classified in the following ways, A 
and B : 

A. Differential equations are classified as ordinary differential 
equations and partial differential equations, according as one, or 
more than one, independent variable is involved. Thus, the equa- 
tions in Ex. 4, Art. 79, and in Exs. 5-8, Art. 80, are partial differen- 
tial equations; the other equations mentioned above are ordinary 
differential equations. (Only ordinary differential equations are 
discussed in this chapter.) 

B. Differential equations are classified as to the order of the 
highest derivative appearing in an equation. Thus, of the exam- 
ples cited above, Equations (2)-(5), Art. 67 (a), are equations of 
the first order; Equations (2), Ex. 5, Art. 68, and (8), Art. 96, are 

394 



217-219.] DIFFERENTIAL EQUATIONS. 395 

equations of the second order; the last equation but one in Ex. 1, 
Art. *>9, is an equation of the nth order. 

A solution (or integral) of a differential equation is a relation 
between the variables which satisfies the equation. Thus, in 
Art. 73, Ex. 1, relation (1) satisfies Equation (4), and, accordingly, 
is a solution of (4). 

Ex. 1. Show that relation (1) satisfies Equation (4) in Art. 73, Ex. 1. 

Ex. 2. See Ex. 4, Art. 79, and Exs. 5-8, Art. 80. In these examples the 
equations in the ordinary functions are solutions of the differential equations 
associated with them. 

Ex. 3. Show that the relations in Exs. 2-5, Art. 73, are solutions of the 
differential equations obtained in these respective exercises. 

218. Constants of integration. General solution. Particular solu- 
tions. It has been seen in Art. 73, Ex. 6, that the elimination of 
n arbitrary constants from a relation between two variables gives 
rise to a differential equation of the nth order. This suggests the 
inference that the most general solution of a differential equation 
of the nth order must contain n arbitrary constants. Eor a proof 
of this, see Diff. Eq., Art. 3, and Appendix, Note C. Simple 
instances of this principle have appeared in Art. 73, Exs. 1-5. 

A general solution of an ordinary differential equation is a solu- 
tion involving n arbitrary constants. These n constants are called 
constants of integration. Particular solutions are obtained from the 
general solution by giving the arbitrary constants of integration 
particular values. The solutions of only a few forms of differential 
equations, even of equations of the first order, can be obtained. 

N.B. Eor a fuller treatment of the topics in Arts. 217, 218, see Diff. Eq., 
Chap. I. 

EQUATIONS OF THE FIRST ORDER. 

219. Equations of the form f(x)dx + F(y)dy = 0. Sometimes 
equations present themselves in this simple form, or are readily 
transformable into it; that is, to use the expression commonly 
used, " the variables are separable." The solution is evidently 



ff(x)dx+JF(y)dy = c. 



396 INTEGRAL CALCULUS. 


[Ch. XXVII. 


Ex. 1. Solve ydx + xdy = 0. 


(1) 


On separating the variables, — + -2 = 0, 

x y 




and integrating, log x + log y = log c ; 




whence xy = c. 


(2) 



Solution (2) can be obtained directly from (1) on noting that ydx + xdy 
is d(xy). 



Ex.2. Vl - x 2 dy + VI - y 2 a*x = 0. Ex.3. w(x + «) dy + w(y + &)<Zx = 0. 

220. Homogeneous equations. These are equations of the form 
Pdx + Qdy = 0, in which P and Q are homogeneous functions 
of the same degree in x and y. TJie substitution of vx for y 
leads to an equation in v and x in which the variables are easily 
separable. 

Ex. 1. (if- - x 2 ) dy + 2 xy dx = 0. Ex. 3. if dx + (xy + x°-) dy = 0. 

Ex. 2. (x 2 + y 2 ) dx + xy dy = 0. Ex. 4. O 2 - 2 sey) dx = (x 2 - 2 xy) dy. 

221. Exact differential equations. These are equations of the 

form 

Pdx+Qdy = 0, (1) 

in which the first member is an exact differential (see Art. 179). 
If P and Q satisfy test (2), Art. 179, then (1) is an exact differ- 
ential equation, and its solution is 



f(Pdx+Qdy) = c. 



Ex. 1. x dy + ydx = 0. (See Ex. 1, Art. 186.) 

Ex. 2. (2 xy + 3) dx + (x 2 + 4 y) dy = 0. 

Ex. 3. (e x sin y -f 2 x) dx + e x cos ydy = % 

Ex. 4. (ax - y 2 ) dy = (x 2 - ay) dx. 

Integrating factors. Equations that are not exact can be made 
exact by means of what are termed integrating factors. In some 
cases these factors are easily discoverable. 



220-222.] DIFFERENTIAL EQUATIONS. 397 

EXAMPLES. 

5. Solve xdy — ydx = 0. (1) 

The first member does not satisfy the test in Art. 179 ; thus (1) is not an 
exact differential equation. Multiplication by 1 -±- xy gives 

dy _ dx _ q . 
y x ~ 

whence log y — log x = log c, and, accordingly, y = ex. 
Multiplication by 1 -r- x' 2 gives 

xdy — ydx __« 

x 2 

V 
whence - = c, i.e. y — ex. 

Similarly, multiplication by 1 h- y 2 makes (1) integrable. 
The multipliers used above are called integrating factors. In the follow- 
ing examples these factors can be obtained by inspection. 

6. Solve (y 2 - x 2 ) dy + 2 xy dx = 0. (See Ex. 1, Art. 220.) 
On rearranging, y 2 dy + 2 xy dx — x 2 dy = 0, 

and using the factor 1 - y 2 , dy + 2 xy dx ~ ^ dy = 0. 

y 2 

x 2 
Whence, on integration, y -\ — = c ; 

y 

i.e. x 2 + y 2 — cy = 0. 

7. 2 ay dx = x{y — a) dy. 8. (y + xy' 2 )dx — (x 2 y — x)dy. 
Note. On Integrating Factors see Biff. Eq., Arts. 14-19. 

222. The linear equation ^4 Py = Q, (1) 

in which P and Q do not involve y. (It is called linear because 
the dependent variable and its derivative appear only in the first 
degree.) This is, perhaps, the most important equation of the 
first order. 

It has been discovered that el pdx is an integrating factor for 
this equation. On using this factor, 

•'"■(f+Ar)-* 1 '*; (2) 

whence, on integration, 

ijei pdx =^Qel pdx dx + c. 

Note. For the discovery of the integrating factor, see Biff. Eq., Art. 20. 



398 INTEGRAL CALCULUS. [Ch. XXVII. 

EXAMPLES. 

1. Show that (2) is an exact differential equation. 

2. x^--ay = x + l. 

dx 

On using form (1), ^ - - y = 1 + x~\ 

dx x 

Here P = --. .-. fp dx = - alogx = log x~ a . ;. el pdx = x~* 

x J 

On using this factor, x~ a (dy — ax~ l dx) = x _a (l + x -1 ) dx ; 



and integrating, 




vx -a - x ~ a+1 + x ~ a + c 

1 — a —a 




whence 




x 1 

y = — ■ 1- cx a . 

1 — a a 




3. (1 - x 2 ) ^ - 
dx 


■xy 


= 1. 4. 


cos 2 x^ 
dx 


Some equations 


are 


reducible to form (1). For 

^ + Py=Qy n . 

dx 


example, 


On division by y 


ire 


y-n <& + pyl-n = £. 

dx 





(3) 



On putting y 1 -" = v, it will be found that (3) takes the linear form 

f x + (l-n)Pv = (l-n)Q. (4) 

6. Derive (4) from (3). 

7 dy + _ry_ = | 8. ^ = x 3 */3 - xy. 

dx 1 — x 2 dx 

223. Equations not of the first degree in the derivative. Three 
types of these equations will be considered here, viz. A, B, C, that 

follow. (Let -^ be denoted by p.) 

A. Equations reducible to the form x = f(y 9 p). (1) 

On taking the ^/-derivatives, -= <l>(y> P, — ] say. (2) 



P V d 2/, 

Possibly, (2) may be solvable and give a relation, say, 

F(p, y, c) = 0. (3) 



222, 223.] DIFFERENTIAL EQUATIONS. 399 

The p-eliminant between (1) and (3) is the solution. If this 
eliminant is not easily obtainable, Equations (1) and (3), taken 
together, may be regarded as the solution, since particular corre- 
sponding values of x and y can be obtained by giving p particular 
values. 

Ex. 1. x = y + a logp. 

On taking the ^-derivative, 1 = l + ^ ^2 ; whence 1 - p = a & • 
p p dy dy 

On integrating, y = c - a log Q> - 1); 

and thence x = c + a log P . 

P 
Ex. 2. p 2 y + 2px = y. Ex. 3. £ = y-f-p 2 . 

B. Equations reducible to the form y =f(x,p). (4) 
On taking the a>derivative, p = <f»fx, p, —) say. (5) 

Possibly, (5) may be solvable and give a relation, say, 

F(p, x, c) = 0. (6) 

The j>eliminant between (4) and (6) is the required solution. 
If this eliminant is not easily obtainable, Equations (4) and (6), 
taken together, may be regarded as the solution, since they suffice 
for the determination of x and y by assigning values to a param- 
eter p. 

Ex. 4. 4 y = x* + p 2 . Ex. 5. 2 y + p 2 = 2 z 2 . 

C. Clairaut's equation, viz. y =poc -\- f(p). (7) 

In this case 2/ = cx +/( c ) (8) 

is obviously a solution. 

This solution can be obtained on treating (7) like (4), of which it is a 
special case. 

Thus, on taking the ^-derivatives in (7), 



P=P + [x+f'(P)l%' 



dp 



From this, s+/'(p) = (9), or ef = - ^ 10 ^ 

Equation (10) gives p = c. 

Substitution of this in (7) gives (8). 

As to the part played by (9) see Diff. Eq. , Art. 34. 



400 INTEGRAL CALCULUS. [Ch. XXVII. 

EXAMPLES. 

6. y=px-\--> 7. y = px + aVl + j? 2 . 

8. x 2 (?/ — px) = yp 2 . [Suggestion : Put x 2 = w, y 2 = v.] 

Note 1. Sometimes the first member of an equation f(x, y, p) = is 
resolvable into factors. In such a case equate each factor to zero, and solve 
the equation thus made. (This is analogous to the method pursued in solv- 
ing rational algebraic equations involving one unknown.) 

9. Solve p 3 - p 2 {x + y + 2) + p(xy + 2 x + 2 y) - 2 xy = 0. 
On factoring, (p — x) = 0, p — ?/ = 0, p — 2 = 0. 

On solving, 2 y = x 2 -f c, y = ce x , 2/ = 2 x + c. 

These solutions may be combined together, 

(2 y - x 2 — c) {y — ce x ) (y — 2 x — c) = 0. 

Note 2. On Equations of the first order which are not of the first degree 
see Diff. Eq. , Chap. III. 

224. Singular solutions. Let a differential equation f(x, y,p)=0 
have a solution f(x, y, c) = 0. The latter is geometrically repre- 
sented by a family of curves. The equation of the envelope of 
this family (Art. 120) is termed the singular solution of the differ- 
ential equation. That the equation of the envelope is a solution 
is evident from the definition of an envelope (see Art. 120) and 
this fact, viz. that at any point on any one of the curves of the 
family the coordinates of the point and the slope of the curve 
satisfy the differential equation. The singular solution is obviously 
distinct from the general solution and from any particular solution. 

For example, the general solution [(8), Art. 223] of Clairaut's equation 
is, geometrically, a family of straight lines. The envelope of this family of 
lines is the singular solution of (7) . The envelope of (8) may be obtained 
by the method shown in Art. 123. Differentiation of the members of (8) 
with respect to c gives — x4-f'( ") 

The envelope is .the c-eliminant between this equation and (8). 

EXAMPLES. 

1. Show that the singular solution of Ex. 6, Art. 190, is y 2 = 4 ax. 

2. Find the singular solutions of the equations in Exs. 7, 8, Art. 223. 



224, 225. J 



DIFFERENTIAL EQ UA TIOXS. 



401 



3. Find the general solution and the singular solution of : 

(1) y=px+p*. (2) p*x = y. (3) 8 a(l +j>) 3 = 27(s + y)(\ -p)*. 

Xote 1. The singular solution can also be derived directly from the dif- 
ferential equation, without finding the general solution ; see reference below. 

Note 2. On Singular Solutions see Diff. Eq., Chap. IV., pages 40-49. 

225. Orthogonal Trajectories. Associated with a family of curves 
(Art. 120), there may be another family whose members intersect 
the members of the first family at right angles. An instance is 
given in Ex. 1. The members of the one family are said to be 
orthogonal trajectories of the other family. 

For example, the orthogonal trajectories of a family of concentric circles 
are the straight lines passing through the common centre of the circles. 



A. To find the orthogonal trajectories of the family 



(1) 



in w T hich a is the arbitrary parameter. Let the differential 
equation of this family, which is obtained by the elimination of 
a (see Art. 73), be 




Fig. 143. 



Fig. 144. 



Let P be any point, through which pass a curve of the family 
and an orthogonal trajectory of the family, as shown in Fig. 143. 
For the moment, for the sake of distinction, let (x, y) denote the 
coordinates of P regarded as a point on the given curve, and let 



402 INTEGRAL CALCULUS. [Ch. XXVII. 

(X, Y) denote the coordinates of P regarded as a point on the 
trajectory. At P the slope of the tangent to the curve and the 

slope of the tangent to the trajectory are respectively -^ and — . 

dx d2L 

Since these tangents are at right angles to each other ? 

dy = dX 
dx dY 

Also x=X, and y=Y. 

Substitution in (2) gives 

+(*> T, -ff)=°- ( 3 ) 

But P(X, Y) is any point on any trajectory. Accordingly, (3) 
or, what is the same equation, 

♦(■"•-SH < 3 ') 

is the differential equation of the orthogonal trajectories of the 
curves (1) or (2). 

Hence: To find the differential equation of the family of orthog- 
onal trajectories of a given family of curves substitute -for — 

in the differential equation of the given family. * 

EXAMPLES. 

1. Find the orthogonal trajectories of the family of circles which pass 
through the origin and have their centres on the x-axis. 
The equation of these circles is 

x 2 + yi = 2 ax, (4) 

in which a is the arbitrary parameter. 

On differentiation and the elimination of a (Art. 73), there is obtained 
the differential equation of the family, viz. 

y i _ X 2 _ 2xy^- = 0. (5) 

dx 

The substitution of - — for ^ gives the differential equation of the 

orthogonal curves, viz. ^ 

y*-x* + 2xy— = 0. (6) 

cly 



225.] 



DIFFERENTIAL EQ UA TIONS. 

r 



403 




Fig. 115. 

Integration of (6) [see Art. 221, Ex. 6] gives 

x 2 + !/ 2 = cy, 



(7) 



the orthogonal family, viz. a family of circles passing through the origin and 
having their centres on the y-axis. (See Fig. 145.) 

2. Obtaiu the orthogonal trajectories of the circles (7), viz. the circles (4). 

3. Derive the equation of the orthogonal trajectories of the family of 
lines y = mx. 

4. Derive the equation of the family of concentric circles whose centre 
is at the origin. 



B. To find the orthogonal trajectories of the family 
f(r, e, c) = 0, 



(8) 



in which c is the arbitrary parameter. Let the differential equa- 
tion of this family, which is obtained by the elimination of c, be 



fU 



dr 

dQ 



(9) 



404 INTEGRAL CALCULUS. [Ch. XXVII. 

Let P be any point through which pass a curve of the given 
family and an orthogonal trajectory of the family, as shown in Fig. 
144. For the moment, for the sake of distinction, let (r, 6) denote 
the coordinates of P regarded as a point on the given curve, and 
let (R, ©) denote the coordinates of P regarded as a point on the 
trajectory. At P (see Art. 63) the tangent to the given curve and 
the tangent to the trajectory make with the radius vector angles 

whose tangents are respectively r — ■ and R 

dr dR 

Since these tangent lines are at right angles to each other, 



™ = L_ ; whence^ = -ri2^ = -rf. 

dr n d®> dd dR dR 



r— = — ; whence — = — rR ^^ = — R 

dR 

Accordingly (9) may be written 



f(r, ©, _j?||) = 0. (10) 

But P(R, ©) is any point on any trajectory. Accordingly (10), 
or the same expression in the usual symbols r and 6, 

is the differential equation of the orthogonal trajectories of the 
curves (8) or (9). 

Hence : To find the differential equation of the family of orthogo- 
nal trajectories of a given family of curves, substitute —r 2 — for — 
in the differential equation of the given family. 



EXAMPLES. 

5. Find the orthogonal trajectories of the set of circles r = acos0, a 

being the parameter. 

Differentiation and the elimination of a gives the differential equation of 

these circles, viz. f i r 

— +r tan.0 = 0. 

dd 

On substituting, as directed above, there is obtained 

r — = tan 6. 
dr 

the differential equation of the orthogonal trajectories. Integration gives 

another family of circles r = c sin 0. (11) 



225.] DIFFERENTIAL EQUATIONS. 405 

6. Sketch the families of circles in Ex. 5, and show that the problem 
and result in Ex. 5 are practically the same as the problem and result in Ex. 1. 

7. Find the orthogonal trajectories of circles (11), viz. the circles in 
Ex. o. 

X.B. Various geometrical problems requiring differential equations are 
given in the following examples. 

Note 1. On applications of differential equations of the first order, see 
Diff. Eq., Chap. Y. 

8. Find the curves respectively orthogonal to each of the following 
families of curves (sketch the curves and their trajectories') : (1) the parabolas 
y 2 = 4 ax ; (2) the hyperbolas xy = k 2 ; (3) the curves a n ~ l y = x n ; interpret 
the cases n — 0, 1, — 1, 2, — 2, ± $, ± §, respectively ; (4) the hypocycloids 

2. 2. _2 

a; 3 -f y 3 = a 3 ; (5) the parabolas y = ax 2 ; (6) the cardioids r = a(l — cos 6) ; 
(7) the curves r n sin nd = a n ; (8) the curves r n = a n cos nd ; (9) the lemnis- 

cates ?* 2 = a' 2 cos 2 ; (10) the confocal and coaxial parabolas r = — ; 

(11) the circles x 2 + y 2 + 2 my = a 2 , in which m is the parameter. ~*~ cos 

9. (a) Show that the differential equation of the confocal parabolas 

y 2 = 4a(x + a) is the same as the differential equation of the orthogonal 

curves, and interpret the result. (5) Show that the differential equation of 

x 2 v 2 

the confocal conies 1 ^ — = 1 is the same as the differential equation 

a 2 + I b 2 + I H 

of the orthogonal curves, and interpret the result. 

10. Find the curve such that the product of the lengths of the perpen- 
diculars drawn from two fixed points to any tangent is constant. 

11. Find the curve such that the product of the lengths of the perpen- 
diculars drawn from two fixed points to any normal is constant. 

12. Find the curve such that the tangent intercepts on the perpendiculars 
to the axis of x at the points (a, 0), (—a, 0), lengths whose product is 6 2 . 

13. Find the curve such that the product of the lengths of the intercepts 
made by any tangent on the coordinate axes, is equal to a constant a 2 . 

14. Find the curve such that the sum of the intercepts made by any 
tangent on the coordinate axes is equal to a constant a. 



EQUATIONS OF THE SECOND AND HIGHER ORDERS. 

Only a very few classes of these equations will be solved here ; 
namely, simple forms of linear equations with constant coefficients 
and homogeneous linear equations. Three special equations of 
the second order will also be briefly discussed. 



406 INTEGRAL CALCULUS. [Ch. XXVII. 

226. Linear Equations. Linear equations are those which are 
of the first degree in the dependent variable and its derivatives. 
The general type of these equations is 

^2/ + P 1 ^M + P 2 ( ?—y+ ••• + P^%L+P l g = X, 
dx n 1 dx n ~ 1 2 dx n ~ 2 x dx 

in which P lf P 2 , •••, P n , X, do not involve y or its derivatives. 

(For some general properties of these equations see Murray, Integral 
Calculus, Art. 113, Biff. Eq., Art. 49.) 

A. The linear equation ^^+Pi^^+P2^— M +^+Pny=0,(l) 
dx n da* 1 - 1 dx n ~ 2 

in which the coefficients P X ,P 2 , ■••, P n , are constants. 

The substitution of e mx for y in the first member, gives 

(m n + P^n"- 1 + P 2 m n ~ 2 -\ \- P n )e mx . 

This expression is zero for all values of m that satisfy the 
equation m „ + p im -i + p^n-2 + . . . + p^ = . ^) 

and, accordingly, for each of these values of m, y = e mx is a solu- 
tion of (1). Equation (2) is called the auxiliary equation. Let 
m x , m 2 , •••, m n , be its roots. Substitution will show that y = c Y e m \ x , 
y = c 2 e m 2 x , • ••, ?/ = c n e m n x , and also 

2/ = c 1 e w i a + c 2 e m P -\ \- c n e m n x , (3) 

in which the c's are arbitrary constants, are solutions of (1). 
Solution (3) contains n arbitrary constants and, accordingly, is the 
general solution. 

Note 1. If two roots of (2) are imaginary, say a + i/3 and a — i/3, i 
denoting V— 1, the corresponding solution is 

y = aela+W* -f- c 2 e(*- f / 3 )*. 

According to Art. 179 this may be put in the form 

y = e<^{c x e^ x + c 2 e- { P x ) 

= e ax {ci(cos fix + i sin /to) + c 2 (cos fix — i sin £#)}, 

= e ax {(ci + c 2 ) cos /?# + i(ci — c 2 ) sin px}, 

= e« x ( J. cos px + B sin /3x) , 

in which A and I? are arbitrary constants, since ci and c 2 are arbitrary 
constants. 



226.] DIFFERENTIAL EQUATIONS. 407 

Note 2. If tiro roots of (2) are equal, say mi and m 2 each equal to a, the 

corresponding solution, viz. 

yi = Cie m i x + c- 2 e m 2 x , 

becomes y = (ci + c 2 )e ax , *.e. ?/ = ce aI , 

which does not involve two arbitrary constants. Put m 2 = a + h ; then the 

solution takes the form , , ,, 

y = ae ax + c 2 e (a+ * )a! , 

= e ax (ci 4- c 2 e hx ). 

On expanding e** in the exponential series (Art. 152, Ex. 7), this equation 
becomes 

y = e ax (A + Bx + \ e 2 h 2 x 2 + terms in ascending powers of h), (4) 

in which A = C\ + c 2 and 2? = c 2 h. On letting ft approach zero in (4), the 

latter becomes . . „ . 

y = e ax (A 4- Ex). 

(The numbers ci and c 2 can always be chosen so that C\ + c 2 and c 2 h are 
finite.) 

If a root a of (2) is repeated >* times, the corresponding solution is 

y = (ci -f c 2 £ + c 3 z 2 + ••• + c r x r - 1 )e aa; . 

Note 3. On Equation (1), see Diff. Eq., Arts. 50-55. 



EXAMPLES. 

1. Solve ^-3^+2y = 0. 

dx* dx y 

The auxiliary equation is m 3 — 3 ra + 2 = ; 
its roots are — 2, 1, 1. 

Accordingly, the solution is y = C\er 2x + (c 2 + CzX)eF. 

2. Solve ^ 4- « 2 ?/ = 0. 

f?x- 2 

The auxiliary equation is ra 2 4- a 2 = 0; 
its roots are ai, — ai. 

Accordingly, its solution is y — c x e aix + c 2 e~ aix 

= A cos ax + B sin ax. (See Ex. 1, Art. 73.) 

3. Solve the following differential equations : 

(1) D 2 y - 4 Dy + 13 y = 0. (2) D 3 ?/ - 7 Z>y + 6 y = 0. 

(3 ) ^_12^-I6y = 0. (4) ^-10^ + 62^-160^+ 136y = 0. 

dx 3 (2z dx* dx* dx 2 dx 



408 INTEGRAL CALCULUS. [Ch. XXVII. 

B. The " homogeneous" linear equation 

in which p 1} p 2 , • ••, p n , are constants. 

First method of solution. If the independent variable x be 
changed to z by means of the relation 

z = log x, i.e. x = e z , 

the equation will be transformed into an equation with constant 
coefficients. (For examples, see Art. 92 and Exs. 3 (i), (v), (vi), 
page 147.) 

4. Show the truth of the statement last made. 

5. Solve Exs. 7 below by this method. 

Second method of solution. The substitution of x m for y in the 
first member of equation (5) gives 

[_m(m — 1) ••• (m — n + 1) +p 1 m(m — 1) ••• (m — n + 2) -\ +p^x m . 

This is zero for all values of m that satisfy the equation 

m(m— l)'"(m— 7i+l)-f-p 1 m(m— !.)••• (m— w+2)H |-p n =0. (6) 

Let the roots of (6) be m lf m 2 , •••, m n ; then it can be shown, 
as in the case of solution (3) and equation (1), that 

y = daf 11 + c 2 x m2 -\ h c n x mn 

is the general solution of equation (5). 

The forms of this solution, when the auxiliary equation (6) 
has repeated roots or imaginary roots, will become apparent on 
solving equation (5) by the first method. 

EXAMPLES. 

6. Show that the solution of (5) corresponding to an r-tuple root m of 
(6), is y — x m [ci + c 2 log x + c 3 (log x) 2 + ••• + c,-(log x) r ~ x ] ; and show that 
the solution of (5) corresponding to two imaginary roots a + ifi, a — 1*/3, of (6) , is 

y = x a [ci cos (j8 log x) + c 2 sin (/3 log »)]. 



226, 227.] DIFFERENTIAL EQUATIONS. 409 

7. Solve the following equations : 

(1) x 2 D 2 y - xDy + 2 y = 0. (2) x 2 D 2 y - xDy + y = 0. 

(3) x 2 Z> : ?/ - 3 xDy + 4 y = 0. (4) x 3 Z>V + 2 x' 2 Z> 2 2/ + 2 y = 0. 

Note 3. Equations of the form 

(« + &*) n ||+J>l(« + ^) n - 2 £S + P2(« + 6x)- 2 |^| + ... +|M , = 

are reducible to the homogeneous linear form, by putting a + bx = z. 

8. Show the truth of the last statement. 

9. Solve (5 + 2x) 2 ^-6(5 + 2x)^+8?/ = 0. 

Note 4. On Equation (5) , see Diff. Eq. , Arts. 65, 66, 71. . 

227. Special equations of the second order. 

A, Equations of the form -j^ 2 -f^y). 

For these equations 2 -^ is an integrating factor. 

EXAMPLES. 

1. ^ + cN = °- ( Se e Ex. 2, Art. 226.) 
dx 2 * v y 

On using the factor 2 ^, 2 ^ ^ = - 2 a 2 ?/ ^. 
dx dx dx 2 dx 

On integrating, ( -j J = — a 2 ?/ 2 + & 

= « 2 (c 2 — y 2 ), on putting a 2 c 2 for &. 

On separating the variables, ' — a dx, 

Vc 2 -j/ 2 

and integrating, sin -1 - = ax -f a. 

This result may be written y = c sin (ax + a) , 

or y = Asin ax + B cos ax. 

2. Show the equivalence of the last two forms. Express A and B in 
terms of c and a, and express c and « in terms of A and _R 

3. Show that 2 -^ is an integrating factor in case A. 

4. Solve the following equations : 

d% — & dx 

(3) If —2 = — =-, find £, given that — = and x = a, when t = 0. 



410 INTEGRAL CALCULUS. [Ch. XXVII 

B. Equations of the form /(^ 5 §|, x\ = 0. (1) 

On letting p denote — , this may be written f[—,p, x )= 0. (2) 

(XX \CIX 

Integration of (2) may give cj>(p, x, c) = 0, 
and this may happen to be integrable. 



EXAMPLES. 

5. Find the curve whose radius of curvature is constant and equal to a. 
(This example is the converse of Art. 99.) 

6. Solve the following equations : 



(2) xWy + Dy = 0. (4) (1 + x)D 2 y + Dy + x = 0. 

C. Equations of the form /(|*|> ||, v) = 0, (1) 

This (see Art. 90) may be written 

f{p%p,y)=o. ( 2) 

Integration of (2) may give 

F(p, y, c) = 0, 
and this may happen to be integrable. 



EXAMPLES. 

7. Solve |*| + a 2 ?/ = 0. (See Ex. 1.) 

This is P-r- = — cC 2 y. 

dy 

Now proceed as in Ex. 1. 

8. Solve the following equations : 

(3) y*IPy + 1 = 0. (4) D*y + (Dy)* + 1 = 0. 

Note 5. For the solution of equations in the form D n y=f(x) 9 see 
Art. 201. 



227.] DIFFERENTIAL EQUATIONS. 411 

Note 6. On forms like A, B, C, see Diff. Eq., Arts. 77, 78, 79, respectively. 

Note 7. References for collateral reading. For a brief treatment of 
differential equations and for interesting practical examples, see Lamb, Cal- 
culus, Chaps. XL, XII. (pp. 456-540) ; also see F. G. Taylor, Calculus, 
Chaps. XXIX.-XXXIV. (pp. 493-564), and Gibson, Calculus, Chap. XX. 
(pp. 424-441). 

EXAMPLES. 

Solve the following equations : 
(1) rdd = tan e dr. (2) (1 + y)dx + x(x + y)dy = 0. 
(3) (4y+3x)dy+(y-2x)dx=0. (4) x^ -y= Vtf+y~ 2 . (5) *+ytana:=l. 

(6) x || - 2y = x* vTT^- (7) (6 x + 4 y + 5)<to + (10 2/ + 4 x + l)eZy = 0. 

dy 4:X 



(S)y(ydx-xdy)+xVW+7>dy = 0. ( 9)£ + -^ y = —— f 

(io) ^fx + x^TT y = $' (11) B -» = ^ ( 12 ) */ 2 = « 2 (l+l> 2 )- 
(13) (px - ?/) (py + x)= h 2 p. (14) p 2 x 3 + x 2 py = 1. (15) x = 2y -Sp 2 . 
(16) p 2 + 2py cot x = y 2 . (17) */Vl + p 2 = a ; also find the singular solution. 
(18) y — px = v 7 6 2 + a 2 _p 2 ; also find the singular solution. (19) xp 2 = (x — a) 2 , 
and also find the singular solution. (20) -^-a 4 y = 0. (21) -| + 4 y = 0- 

(22)^-3^ + 4^0. (23)x 2 ^ + ^- y = 0. 

»*» <w(3)'4: <»'-S+2^ <*>S)-(8 ,: 



APPENDIX. 

NOTE A. 

ON HYPERBOLIC FUNCTIONS. 

1. This note gives a short account of hyperbolic functions and 
their properties. The student will probably meet these functions 
in his reading ; for many results in pure and applied mathematics 
can be expressed in terms of them, and their values are tabulated 
for certain ranges of numbers.* There are close analogies between 
the hyperbolic functions and the circular (or trigonometric) func- 
tions (a) in their algebraic definitions, (6) in their connection with 
certain integrals, (c) in their respective relations to the rectangular 
hyperbola and the circle. 

2. Names, symbols, and algebraic definitions of the hyperbolic 
functions. The hyperbolic functions of a number x are its hyper- 
bolic sine, hyperbolic cosine, hyperbolic tangent, •••, hyperbolic 
cosecant, and the corresponding six inverse functions. These func- 
tions have been respectively denoted by the symbols sinh x, cosh x, 
tank x, coth x, secli x, cosech x, sink' 1 x, etc. These are the symbols 
in common use. As to symbols for the hyperbolic functions, the 
following suggestion has been made by Professor George M. 
Minchin in Nature, Vol. 65 (April 10, 1902), page 531: "If the 
prefix hy were put to each of the trigonometrical functions, all the 
names would be pronounceable and not too long. Thus, hysinx, 
hytanx, etc., would at once be pronounceable and indicate the 

* See tables of the hyperbolic functions of numbers in Peirce, Short Table 
of Integrals (revised edition, 1902), pages 120-123; Lamb, Calculus, Table 
E, page 611 ; Merriman and Woodward, Higher Mathematics, pages 162-168. 

413 



414 INTEGRAL CALCULUS. 

hyperbolic nature of the functions." This notation will be 
adopted in this note.* 

The direct hyperbolic functions are algebraically defined as follows : 

hysin oc = - — -H — , hycos oc = ^ — ? 

hytena ? =^^ = ea> - 6 " a? , hycot^ = te^ = ** + g % (1) 

hycos a? #* + e-» hysin a? e* - e-* v ' 

1 -i 

hysec a? = - — - — ? hycosec as 



a? hysin a? 

There is evidently a close analogy between these definitions 
and the definitions and properties of the circular functions. [See 
the exponential expressions (or definitions) for sin x and cos x in 
Art. 153.] 

From the definitions for hysin x and hycos x can be deduced, by 
means of the expansions for e x and e~ x (see Art. 152, Ex. 7), the 
following series, which are analogous to the series for since and 
cos x (Art. 152, Exs. 2, 5) : 

hysin a3 = a; + ^ + f^ + .... 

8 ' 5 ' (2) 

hycosx = l+|? + |i + ... 5 

The second members in equations (2) may be regarded as defi- 
nitions of hysin x and hycos x. 

EXAMPLES. 

1. Derive the following relations, both from the exponential defini- 
tions of sin a;, cos a;, hysin x, hycos as, and from the expansions of these func- 
tions in series : (1) cos x = hycos (ix) ; (2) i since = hysin (ix) ; (3) cos (ix) 
= hycos x ; (4) sin (ix) = i hysin x. 

2. (a) Show that e x = hycos x + hysin x, e~ x = hycos x — hysin x. 
[Compare Art. 179 (1), (2).] (b) Show that hysin = 0, hycos = 1, 
hytan0 = 0, hysin oo = oo, hycosoo = co, hytan co = 1, hysin ( — x) = 
— hysin as, hycos ( — x) = hycos x, hytan ( — x) = — hytan x. 

* The symbols used in W. B. Smith's Infinitesimal Analysis are hs, he. 
Jit, hct, hsc, hesc. 



APPENDIX. 415 

3. Show that the following relations exist between the hyperbolic 
functions : 

(1) hycos 2 x — hysin 2 x = 1 ; (2) hysec 2 x -f- hytan 2 x = 1 ; 

(3) hysin (x ± y) — hysin x • hycos y ± hycos x • hysin y • 

(4) hycos (x ±y)= hycos x • hycos y ± hysin x • hysin y ; 

(5) hytan (x ± y) — (hytan x ± hytan y) -> (1 ± hytan x • hytan y) ; 

(6) hysin 2 x = 2 hysin x • hycos x ; 

(7) hycos 2 x = hycos 2 x + hysin 2 x = 2 hycos 2 x — 1 = 1 + 2 hysin 2 x ; 

(8) hytan 2 x = 2 hytan x -4- (1 + hytan 2 x). 

Compare these relations with the corresponding relations between the 
circular functions. 

4. Show the following: (1) *&3***1 = hycos x ; (2) d ( h r cos x) = 

dx dx 

bysinx; (3) ^ b >' tan *) =hysec 2 x ; (4) ^cotx)^ _ h 2 (5) ^(hysecx) 
dx dx dx 

= - hysec x ■ hytan x ; (6) ^ hycsc ^ = - hycsc x • hycot x ; (7) f hysin x dx 

dx J 

= hycos x ; (8) I hycos x dx = hysin x ; (9) ( hytan x dx = log (hycos x) ; 
(10) \ hycot x dx = log (hysin x) ; (11) l hysec x dx = 2 tan-V ; 

(12) ( hycsc xdx = log (hytan -]. Compare these relations with the cor- 
responding relations between the circular functions. 

5. Make graphs of the functions hysin x, hycos x, hytan x. (See Lamb, 
Calculus, pp. 42, 43.) 

V X X 

6. Show that the slope of the catenary - = hycos - is hysin -• Sketch 
... a a a 

this curve. 

Inverse hyperbolic functions. The statement "the hyperbolic 
sine of y is x" is equivalent to the statement "y is a number 
whose hyperbolic sine is x." These statements are expressed in 
mathematical shorthand, 

hysin y = x, y = hysin-i x. (3) 

The last symbol is read " the inverse hyperbolic sine of x" or 
"the anti-hyperbolic sine of xP The other inverse hyperbolic 
functions are defined and symbolised in a similar manner. 

The inverse hyperbolic functions can also be expressed in terms 
of logarithmic functions, and thus they may be given logarithmic 
definitions. (This might have been expected, for the direct hyper- 
bolic functions are defined in terms of exponential functions, and 
the logarithm is the inverse of the exponential.) 



416 INTEGRAL CALCULUS. 

Let hysin y = x ; then x = \{e y — e~ y ). 

This equation reduces to e 2y — 2 xe y — 1 = 0. 

On solving for e v , e y = x + V^ 2 + 1. (4) 

(For real values of y, e y being positive, the positive sign of the 
radical must be taken.) 



From (4) y = hysin-i x = log(x + Va;2 + 1). (5) 

N.B. The base of the logarithms in this note is e. 

In a similar manner, on putting 

x = hycos 2/ = \ (e y + e~ y ), 
and solving for e y , 



e y = x± Vx 2 - 1. (6) 

For real values of y, x is greater than 1 and both signs of the 
radical can be taken. 

From (6) and the fact that (x + V# 2 — l)(x— Vx 2 — 1) = 1, and 
thus log (x — V# 2 — 1) = — log (x + V& 2 — 1), it follows that 



y = hycos-i x = ±log(x + Vx* - 1). (7) 

In a similar manner it can be shown that 

hytan-i a5 =|logli|, (8) 

where x 2 < 1 for real values of hytan -1 a: ; and that 

hycot-i^ = |log^±|, (9) 

where x 2 > 1 for real values of hycot -1 a?. 



EXAMPLES. 

7. Derive the relations (7), (8), (9). 

8. Solve equations (5), (7), (8), (9), for x in terms of y, and thus 
obtain the definitions of the direct hyperbolic functions. 

9. Show that the differentials of hysin -1 x, hycos -1 x, hytan -1 x, hycot -1 x, 

are respectively dx , ± dx , — — for x 2 < 1, — for x 2 > 1. 

Vx 2 + 1 Vx 2 - 1 1 - x 2 x 2 - 1 

Compare these with the differentials of sin -1 x, cos -1 x, tan -1 x, cot" 1 x. 



APPENDIX. 417 

10. Following the method by which relations (5) -(9) have been derived, 
show that : 



hysin-^ = log a; + Va;2 + a2 ; hycos-i g = ± log ' 8 ± Vx2 ~ a2 ; 
a a a a 

hytan-l - = \ log ^^ for x 2 < a 2 ; hycot-i - = 1 log ^^ for x 2 > a 2 . 
a 2 a — a? a 2 x — a 

11. Assuming the relations in Ex. 10, show that the x-diff erentials are : 

d( hysin-i ^ = ^ x ; d( hycos-i *\ = ± — ^ ; 

V a) Vx 2 + a 2 V «/ Vx 2 -a 2 

d(hytan-i^\=-^- f rx 2 <a 2 ; dfhycofi - W - a rte for x 2 >a 2 . 

V a/ a 2 - x 2 \ a) x 2 — a 2 

Compare these differentials with the differentials of sin -1 -, cos -1 -, tan -1 -, 

a a a 

cot-i?. 
a 

12. Assuming the relations in Ex. 10 as definitions of the inverse hyper- 
bolic functions, derive the definitions of the corresponding direct hyperbolic 
functions. (Suggestion. Follow the plan outlined in Ex. 8.) 

3. Inverse hyperbolic functions defined as integrals. It follows 
from Ex. 11. Art. 2, that 

C dx = hysin- 1 - + c ; f dx = ± hycos- 1 - + c ; 
J Vz 2 + a 2 a J Vz 2 - a 2 a 

/dx 1, , _-. x . / 9 ^ 9 N r dx li . _x a? . „ 
= -hytan * - + c ,(x 2 <cr); - = --hycot Y -+c, 
a- — x- a a J x~—a a a 

(z 2 >a 2 ). 

Accordingly, these inverse hyperbolic functions can be ex- 
pressed in terms of certain definite integrals, viz. : 

f-_jte_ = j u + Vg±g = in _ x u 

Jo Vic2 + a 2 a a' 

C' -JS— = log «±v^E« , = ± hycos -i «. 
J« Vrr2-a2 a a' 

r»_j§2_ * , a±» = i h ,« ui<a , 

Jo a* -X2 2 a a — u a a 

C«d^_ _ 1 ! u + a = _i hycot -i»?, W 2>a2. 
J x sc* -a* 2a. u — a a a' 



418 INTEGRAL CALCULUS. 

These relations between definite integrals and inverse hyperbolic 
functions may be taken as definitions of the functions. 

The inverse circular functions can be defined by integrals which 
are very similar to the integrals appearing in the definitions of the 
hyperbolic functions. Thus : 



J dx 
o Va^" 

s 



dx 
sin x -, = — cos" 



s. 



Va 2 — x 2 



dx =ltan->™ f dx =-lcot-'?, 



o a 2 + x 2 a a J«> a 2 + x 2 a 



EXAMPLES. 

1. Find the area of the sector AOP of the hyperbola x 2 — y 2 = a 2 
(Fig. 147), P being the point for which x = u. Thence show, from the 
definition above, that hycos -1 - is the ratio of twice the sector AOP to the 
square whose side is a. 

2. Find the area of the sector BOP' bounded by the y-axis, the arc 
BP' of the hyperbola y 2 — x 2 = a 2 (the conjugate of the hyperbola in Ex. 1), 
and the line OP' drawn from the origin to the point P , P 1 being the point for 
which x — u. Then show, from the definition above, that hysiir 1 — is 
the ratio of tivice the sector BOP' to the square whose side is a. 

3. Sketch the curve y{a 2 — x 2 ) =a 3 . Calculate the area between this curve, 
the axes, and the ordinate for which x=u(u 2 <a 2 ). Show that hytarT 1 - is the 
ratio of this area to the area of the square whose side is a. 

4. Sketch the curve y(x 2 — a 2 ) = a 3 . Calculate the area bounded by this 
curve, the x-axis, and the ordinate at x = u(u 2 >a 2 ). Show that hycot -1 - is 
the ratio of this area to the area of the square whose side is a. 

4. Geometrical relations and definitions of the hyperbolic functions. 

In Fig. 146 P is any point (x, y) on a circle x 2 + y 2 = a 2 . Let the 
area of the sector A OP be denoted by u and the angle AOP by 0. 
Then, by plane trigonometry, 

u = \a 2 0; whence, 0-^|- (1) 

In Fig. 147 P is any point on a rectangular hyperbola x 2 —y 2 =a 2 . 
(The a of the hyperbola bears no relation whatever to the a of 



APPENDIX. 



419 



the circle.) Let the area of the sector AOP be denoted by u. 
Then 



« = area OPM— area APM = i xy — I V^ 



a 2 dx: 



whence, u = % log ^+V^-« 2 = t log «±£t 

^ CI ^ (X 



(2) 



From (2), log - + ^ = — ^ ; whence, 
a- a 2 



Also, since x 2 — y 2 = a 2 , 



x + y 
a 

x — y 
a 



e a . 



= e 



(3) 




x o 

Fig. 146. Fig. 147. 

From equations (3), on addition and subtraction, 



2u _2u 2u _2u -j 



_2m 

e~« 2 



6 a 2 _|_ g a 2 



(4) 



* That is, !< = | a 2 hycos" 1 - ; whence, hycos- 1 - = — . 
a a a 2 

t If a = 1, log (x 4- y) = 2 i< = twice area ^LOP. On account of the relation 
between natural logarithms (i.e. logarithms to base e) and the areas of hyper- 
bolic sectors, natural logarithms came to be called hyperbolic logarithms. 
The connection between these logarithms and sectors was discovered by 
Gregory St. Vincent (1584-1667) in 1647- 



420 INTEGRAL CALCULUS. 

Eelations (4) lead to geometrical definitions of the hyperbolic func- 
tions. These definitions are given in the following scheme. This 
scheme, supplemented by relation (1), also shows the close geo- 
metrical analogies existing between the hyperbolic and the circular 
functions. 

N.B. In Figs. 146, 147 the a and u of the circle are not related 
in any way to the a and u of the hyperbola. 

In a circle x? + y 2 = a 2 (Fig. In a hyperbola x 2 — y 2 = a 2 

146), if P is any point {x, y) (Fig. 147), if P is any point 
and u = area sector AOP, (x, y) and w = area sector AOP, 



then 


y ■ 2u 

~ = sm — > 
a a 1 

x 2 u 

- = cos -p 

a a' 






then 


^hysin^, 


whence, 


— — tan — - , 
x a? 






whence. 


Jo^lS, 


2u 
a 2 


r 1 ^-= cos - = 
a a 


= tan _ 


X 


2u_ 

«2 


hysin- 1 ^ = hyc 



a 

= hytan-i £. 

a? 

These results may be expressed in words : 

The circular functions may be defined by means of the relations 
connecting a point (x, y) on the circle x 2 -\-y 2 = a 2 and a certain cor- 
responding circular sector; and the hyperbolic functions may be 
defined by means of the relations connecting a point (x, y) on the 
rectangular hyperbola x 2 — y 2 = a 2 and a certain corresponding hyper- 
bolic sector. 

Each of the inverse circular functions may be expressed as the ratio 
of twice the area of a certain sector of a circle of radius a to the 
square described on the radius of the circle, and each of the inverse 
hyperbolic functions may be expressed as the ratio of twice the area of 
a certain sector of a rectangular hyperbola of semi-axis a to the square 
described on this semi-axis. 

(For a more general notion see Ex. 3 following.) 
The term hyperbolic arose out of the connection of these func- 
tions with the hyperbola. 



APPENDIX. 421 



EXAMPLES. 

1. Show that hysin -1 f == hycos -1 f = hytan -1 f . Represent each of these 
functions geometrically. Compute hysin -1 f. \Ans. 1.099.] 

2. Show that hysin -1 f = hycos -1 f = hytan -1 f. Represent each of these 
functions geometrically. Compute hysin -1 f . [Ans. .693.] 

3. Show that, if AP (Fig. 146) is an arc of an ellipse b 2 x 2 + a 2 y 2 = a 2 b 2 , 
and u denote the area of the elliptic sector AOP, it is possible to write 

* = cos^, y = sm^. 
a ab b ab 

Also show that, if AP (Fig. 147) is an arc of a hyperbola — — *- = 1, and 
u denote the area of the hyperbolic sector AOP, then a 



u =rt\og(*±y\- 



2 ~\a b 

and thence show that 

* = hycos 2«, y= hysin *« 

a. a& 6 a& 

(Williamson, Integral Calculus, Arts. 130, 130 a.) 

4. Show that a point P(x, z/) on the ellipse -^ + ^ — 1 m Ex. 3 may be 
represented as (a cos 0, &sin0), and show that 0(= eccentric angle of P) 

= (2 area sector A OP + ab). „ 

x 2 y 2 
Show that a point P(x, ?/) on the hyperbola — — ^ = 1 in Ex. 3 may be 

represented as (a hycos v, & hysin v), and show that v =( 2 area sector 
AOP+ab). ' 

5. The Gudermannian. Suppose that 

sec <f> + tan cf> = hycos v + hysin v. (1) 

From (1) and the identities sec 2 <f> — tan 2 <£ = 1, hycos 2 v — 
hysin 2 v = l, it follows that 

sec <j> = hycos v, (2) tan <£ = hysin v. (3) 

Since [see Art. 2, Ex. 2 (a)] log (hycos v + hysin v) = v, relation 
(1) may be written v = ^ (sec + + ^ +) . (4) 

that is, by. trigonometry, 

= log tan (! + *) = 2.302585 log 10 tan f| + |Y (5) 



422 INTEGRAL CALCULUS. 

When any one of the relations (l)-(5) holds between two numbers 
v and <f>, </> is said to be the Gudermannian of v.* This is expressed 
by this notation: + = gdv . (6) 

In accordance with the usual style of inverse notation each of 
the relations (4), (5), (6) is expressed 

v = gd-i$. (?) 

The second members of (4) and (5) are more frequently denoted 
by the symbol \(4>), which is read " lambda <j>," than by gd' 1 cf>. 

Geometrical representation of A(<£) or gdr 1 $. If at P(x, y) in 
Fig. 147, x = a sec cf>, then y = a tan <f>, since x 2 — y 2 = a 2 . On mak- 
ing this substitution for x and y, it can be deduced that 

area sector AOP —\a 2 log (sec $ + tan <f>). (8) 

From this, 

log (sec <f> -f tan <£), i.e. \ (<f>) (or gdr Y cj>) = — - (9) 

a 

From (4), (6), (8), 4> = gd( ^ ' sect ° r A0P y (10) 

If the area of sector AOP be denoted by u, relations (9) and 
(10) may be expressed 

._i , 2 u , ,2u 
gd 1 <£ = — , <j> = gd — . 
a 2 a 2 

To construct an angle whose radian measure is <f>. In Fig. 147, 
about as a centre with a radius a describe a circle. From M 
draw a tangent to this circle, and let the point of contact be at P' 
in the first quadrant ; and draw OP'. Now OM= OP 1 sec MOP ; 
i.e. x = a sec MOP. But, according to the hypothesis in the last 
paragraph, x = a sec <£. Hence, angle MOP' = <|>. 

If a point P(as, y) on the hyperbola x 2 — y 2 = a 2 (see Ex. 4, Art. 4) be 
denoted as (asec0, a tan 0), is the angle which has just now been con- 
structed. 

* This name was given by the great English mathematician Arthur Cayley 
(1821-1895) "in honour of the German mathematician Gudermann (1798- 
1852), to whom the introduction of the hyperbolic functions into modern 
analytical practice is largely due." (Chrystal, Algebra, Vol. II., page 288.) 



APPENDIX. 423 



EXAMPLES. 

1. Derive result (8). 

2. («) Show that, and v being as in equations (l)-(7), 

gdv = sec -1 (hycos u) = tan -1 (hysin v) = cos -1 (hysec v) = sin -1 (hytan v) 

= cot -1 (hycosec v) = eosec -1 (hycot v) ; hytan % v = tan » <|>. 
(6) Show that gd~ 1 <p = hycos -1 (sec 0)= hysin -1 (tan 0); ^dx=2tan- 1 e x — -• 

3. (a) Show that the derivative of X(0)(i.e. gd' 1 ^) is sec0. (6) Show 
that X(-0) = — X(0). [Suggestion. Show that X(— 0)+ X(0) = logl.] 
(c) Sketch the graph of X(0). 

4. Show that ( hysec w ch* = gd u ; ( sec u du = gd* 1 u. 

Note. References for collateral reading on hyperbolic functions. Gib- 
son, Calculus, §§ $<o, 111, 116 ; Lamb, Calculus, Arts. 19, 23, 40, 44, 72, 98, 
Exs. 2, 3 ; F. G. Taylor, Calculus, Arts. 62-80, 439 ; W. B. Smith, Infinitesi- 
mal Analysis, Vol. I., Arts. 99-113; McMahon and Snyder, Biff. Cal., 
pp. 320-325. For further information see Chrystal, Algebra (ed. 1889), 
Vol. II., Chap. XXIX., §§ 24-31 (pages 276-291) ; the notes on pages 288, 
289 contain interesting information about the history and literature of the 
subject. Also see Hobson, Treatise on Plane Trigonometry, Chap. XVI. 
An excellent account of hyperbolic functions, starting from the geometrical 
standpoint and showing practical applications, is given in McMahon, Hyper- 
bolic Functions {i.e. Merriman and Woodward, Higher Mathematics, Chap. 
IV, pages 107-168). 

NOTE B. 

INTRINSIC EQUATION OF A CURVE. 

1. The intrinsic equation of a curve. Usually the equation of a 
curve involves either the Cartesian coordinates x and y or the 
polar coordinates r and 6. In some cases the intrinsic equation 
is especially useful. In the intrinsic equation of a curve the 
coordinates chosen for any point P are (a) the distance of P from 
a chosen fixed point on the curve, this distance being measured 
along the curve, and (b) the angle made by the tangent at P with a 
chosen fixed tangent of the curve. These coordinates are denoted 
respectively by s and <j>. The relation connecting them, f(s, <£)=0 
say, is called the intrinsic equation of the curve. The term 
intrinsic is used because the coordinates s and <f> are independent 
of all points or lines of reference other than the points and 
tangents of the curve itself. 



424 



INTEGRAL CALCULUS. 



EXAMPLES. 

1. Derive the intrinsic equation of a straight line. Let AB be any 

straight line. Let be the chosen 

, j j fixed point, and P(s, 0) be any point 

on the line. It is required to find the 
equation which is satisfied by s and 0. 

The direction of the line at P is the same as the direction at ; hence the 
intrinsic equation is = 0. 

rT 



2. Derive the intrinsic equa- 
tion of a circle of radius a. 
Take (Fig. 107) for the fixed 
point, and the tangent at for 
the tangent of reference. Let 
P(s, 0) be any point on the 
circle. Then s = arc OP and 
= angle TBP. Now arc OP 
= a • angle ; i.e. s = a<j>. 



P(s,<» 



£z! 




Fig. 148. 



2. Derivation of the intrinsic equation of a curve. The intrinsic 
equation of a curve is usually derived from its equation in 

Cartesian coordinates or from its 
equation in polar coordinates. The 
general method of doing this will 
now be shown. 

Let the equation of the curve be 

f(x,y) = 0. (1) 

Take Q for the fixed point, and 
the tangent at for the tangent of 
reference. Take any point P on the 
curve ; let its Cartesian coordinates 
be x, y, and its intrinsic coordinates be s, <£. 
Express s in terms of x, y ; suppose that 

s=Mx,y). (2) 

Also express <£ in terms of x, y ; suppose that 

4>=f 2 (x,y). (3) 

The elimination of x and y between equations (1), (2), (3), will 
give the required equation between s and <£. 




Fig. 149. 



APPENDIX. 425 

Similarly, let the polar coordinates of P be r and 0, and let 
the polar equation of the curve be 

F(r,d) = 0. (4) 

Express s in terms of r, ; suppose that 

s = F 1 (r,0). (5) 

Also express </> in terms of r, ; suppose that 

4> = F 2 (r,0). (6) 

The elimination of r and between equations (4), (5), (6), will 
give the required equation between s and <£. 

Note. A tangent parallel to the x-axis is usually chosen for the tangent 
of reference. 

EXAMPLES. 

1. Derive the intrinsic equation of the hypocycloid 

x* + y* = a*. (1) 

Take the cusp on the positive part of the x-axis for the fixed point, and 
the tangent there for the tangent of reference. Then at any point P(x, y) 
on the arc in the first quadrant 

tan = -0/3 ^ai), (2) 

and « = f aJ (a*-x*). (3) 

From (1) and (2), sec 2 <p = tan 2 + 1 = a? + x%. 

2 

Substitution for x 3 in (3) gives 2 s = 3 a sin 2 0. 

2. If in Ex. 1 the chosen fixed point be at a distance b along the 
curve from the cusp and the chosen fixed tangent (not necessarily at 0) 
make an angle a with the tangent at the cusp, show that the intrinsic 
equation of the hypocycloid is 

2 (s+ b) =3asin 2 (0 + a). 

3. Find the intrinsic equation of the cardioid r = a(l — cos 6). 

Let the polar origin be chosen for the fixed point, and the tangent there 
be chosen for the tangent of reference. Let P(x, y) be any point on the 

cardioid. Then s = f\'*' 2 + l—Ydd = 4 all - cos-V (1) 



426 INTEGRAL CALCULUS. 

Also, (Art. 63) , = 6 + tan" 1 7 -^-=d + tan" 1 [tan -\ = %0. (2) 

On substituting in (1) the value of 6 from (2), 

s = 4a( I — cos — )• 

4. If in Ex. 3 the chosen fixed point be at a distance b from the polar 
origin and the chosen tangent of reference make an angle a with the tan- 
gent at the polar origin, show that the intrinsic equation of the cardioid is 



= 4a[] 



l_cos^+^ 



5. Derive the intrinsic equation of each of the following curves, the 
fixed point and the fixed tangent being as indicated : (1) the catenary 

X X 

y = - (e a + e *), the vertex and tangent thereat ; (2) the parabola y 2 = 4 ax, 

6 
the vertex and tangent thereat ; (3) the parabola r = a sec 2 -, as in (2) ; 

(4) the cycloid x = a(d — sin 6), y = a(l — cos 6), with reference to (a) 
the origin and tangent thereat, (b) the vertex and tangent thereat ; (5) the 
logarithmic spiral r = ce ae ; (6) the semi-cubical parabola Say 2 = 2 x 3 , the 

origin and tangent thereat ; (7) the curve y — a log sec -, the origin ; 

a 

(8) the semi-cubical parabola y % = ax 2 ; (9) the tractrix x = Vc 2 — y 2 + 
clog^-± — G ~ ~ y } the point (0, c). (For an account of the tractrix and 

y 

for various problems which reveal its properties, see the text-books of 
Williamson, Byerly, Lamb, and F. G. Taylor, on the calculus.) 

[Answers : Ex. 5. (1) s = a tan 0, (2) s = a tan <f> sec + a log tan 

(£ + -V (3) as in (2), (4) fa) s = 4 o(l - cos 0), (b) s = 4asin0, 
\2 4/ /. x 

(5) s = c(e«* - 1), (6) 9s = 4a(sec 3 0- 1), (7) s = alogtan [2 + -), 
(8) 27s=8a(sec 3 - 1), (9) s = clogsec 0.] ^ 2 4 ^ 



3. Radius of curvature derived from the intrinsic equation. The 

radius of curvature at a point on a curve can very easily be 
deduced from the intrinsic equation. For, according to Arts. 98, 
99, the radius of curvature being denoted by R, 



APPENDIX. 427 

EXAMPLES. 

1. In Art. 2, Ex. 5 (1), B = a sec 2 0. 

2. Find the radius of curvature for each of the curves in Art. 2, Ex. 1, 
Ex.3, Ex. 5 (4), (5), (6), (9). 

[Answers : Ex. 1. f a sin 2 ; Ex. 3. f a sin ^ ; Ex. 5 (4). (a) 4 a sin 0, 

o 
(6) 4 a cos ; (5) a ce a * ; (6) fa sec 3 tan ; (9) c tan 0.] 

Note. On the intrinsic equation of a curve, see Todhunter, Integral 
Calculus, Arts. 103-119 ; Byerly, Integral Calcuhis, Arts. 114-123. 



NOTE C. 

LENGTH OF A CURVE IN SPACE. 

(This note is supplementary to Arts. 209, 210.) 

The lengths of plane curves have been derived in Arts. 209, 
210. The principle used there is that the length of an arc is the 
limit of the sum of the lengths of infinitesimal chords inscribed 
in the arc. The same principle is employed in finding the lengths 
of curves in space. 

Thus in Fig. 93 or Fig. 95, 

limit of the sum of chords PQ, inscribed from 



length of arc AB = 

1 A to B, when the chords approach zero. 



Now length of chord PQ = V (Ax) 2 + (Ay) 2 + (Az) 2 (1) 



v 



-i2T + @J- <2 > 



Hence, by the definitions in Arts. 22, 166, 

x at B 

length of are AB =J\/l + (|J + (|J fa (3) 

x at A 

Similarly there can be derived from (1), 



y at 

2 



length of arc AB = f^l+f*?Y+ 



y at A 

z&t i 

= f\ 

at A 



/dz 
dyj ' V<fy 



dy 



j\Mfy+(sy- <«> 



428 



INTEGRAL CALCULUS. 



If the coordinates (x, y, z) are expressed in terms of a third 
variable, t say (e.g. see Arts. 158, 159), relations (1), (2) can be 
expressed thus : 

length of chord PQ=J^Y + (f Y +(£)**; (o) 



W 



whence, length of arc ^= j\/(f Y+(f Y+f 






eft. 



(6) 



EXAMPLES. 

1. Find the length of the helix, a curve traced on a right circular cylinder 
so as to cut all the generating lines (elements) of the cylinder at the same 
angle. 

The equations of the helix, as derived below, are 

x = acos6, y = asmd, z = adt&na, (1) 

in which a is the radius of the right circular cylinder x 2 + y 2 = a 2 , and a is 
the angle at which the helix cuts the elements of the cylinder. 
Equations (1) may be written 

x = a cos 0, y = a sin 0, z — c0, (2) 

in which c = a tan a. 

z\ 




Fig. 151. 



In Fig. 150 P(x. y, z) is any point on the helix. 



APPENDIX. 



429 



Fig. 151 shows the cylindrical surface ACB "unwound" and laid out as 

a plane surface. At P : 

x = On — a cos 0, 

y =z nm = a sin 0, 

z = Pm = Am tan a (Figs. 150, 151), 

= ad tan cc. 

The length of the arc APB (Fig. 150) = length of the straight line APB 
(Fig. 151) = Am C sec a = ira sec a. 

Accordingly, the length of the arc which encircles the cylinder = 2 wa sec a. 
This length s will now also he derived by the calculus method shown in this 
article. 

From equations (1) on differentiation, 

— = - a sin d, & = a cos d, — = a tan a. 
dd dd dd 



*• — fvsr+d^^v- 



<Z0/ 



= 21"' Va' 2 sin 2 + a 2 cos 2 6 + a 2 tan 2 cc d0 

= 2aVl + tan 2 « y dd = 2ira sec a. 

Thus, if a = 10 inches and a = 30°, the length of an arc encircling the 
cylinder is 72.5 inches. 

2. Show that the length of the arc of the helix in Ex. 1 from Q — Q x to 
8 = d 2 is 2 a(9 2 — #i) sec a. Hence find that the length of the arc on a cylin- 
der of radius 4 inches from d = 25° to d = 75° when a = 35° is 8.6 inches. 

3. Show that the equations (2) of the helix in Ex. 1 can be transformed 

into x 2 + y 2 = a 2 , y = x tan - ■ 

c 

Then, using these equations, find the length of the arc encircling the 
cylinder. 

4. Show that the equations (2) of the helix in Ex. 1 can be transformed 

into x — a cos - , y = asm-- 

c c 

Then, using these equations, show that the length of the arc measured 
from the point where z = to the point where z = z\ is 



Vn* 



430 INTEGRAL CALCULUS. 

5. Show that the length of the arc of the curve 

x = 2 a cos t, y = 2 a sin t, z = bt 2 , 
from the point at which t = to the point at which t = t\ is 



-* Vqs + W + -^ log & * 1+ Va2 + bHl • 

2 2 6 a 

Sketch the curve. 

6. Show that the length of the arc from the point on the x?/-plane to the 
point where x — x± on the curve 

£-£ = 1, » = l4(a5+a-5), is (* + *Q*Vi?=3f. 

a' 2 & 2 2 a 

Make a figure showing this curve. 

7. Show that the length of an arc of the curve 

x = 4 a cos 3 0, y = 4 a sin" 0, s = 3 c cos 2 0, 
from the point at which 6 = a to the point at which d = /3 is 
3 Va' 2 + c 2 (cos2« - cos 2/8). 

2. 2. 2 

Show that this is a curve encircling the cylindrical surface x 3 + y 3 =(4a) 3 . 
Make a figure with a sketch of the curve, and show that its length is 24 Va' 2 + c' 2 . 



QUESTIONS AND EXERCISES FOR PRACTICE 
AND REVIEW. 



>**c 



A large number of examples are contained in several works on 
calculus, in particular in those of Todhunter, Williamson, Lamb, 
Gibson, F. G. Taylor, and Echols. Special mention may also be 
made of Byerly's Problems in Differential Calculus (G-inn & Co.). 
Exercises of a practical and technical character, which are con- 
cerned with mechanics, electricity, physics, and chemistry, will 
be found in Perry, Calculus for Engineers (E. Arnold) ; Young 
and Linebarger, Elements of the Differential and Integral Calculus 
(D. Appleton & Co.) ; Mellor, Higher Mathematics for Students of 
Chemistry and Physics (Longmans, Green & Co.). Many of the 
following examples have been taken from the examination papers 
of various colleges and universities. 

CHAPTERS II., III., IV. 

1. Explain what is meant by a continuous function. 

2. Explain what is meant by a discontinuous function. Give examples. 

3. (1) Given that f(x) = x 2 + 2 and F(x) = 4 + Vx, calculate f{F(x)} 
and F{f(x)}. (2) If f(x) = x —l, s how that /(*) ~ /M = X ~V . (3) if 

2 + sV X + l l+/(*)/G0 1 + a* 

y = f(x) = — ^ — - and z=f(y), calculate 2 as a function of x. (4) If 
4 - 7 x 

2 x — 1 
y = (f>(x) — , show that x = 0(y), and show that x = 2 (x), in which 

3x — 2 -j 

<P' 2 (x) is used to denote <p{<p(x)}, not {(j>(x)} 2 . (5) If f(x) = , show 

x — 1 

that f 2 (x) = x, fHx) = f(x),fHx) = x. (6) If y =f(x) = ax + h , show that 

ex — a 

* = /(*/)• (") If /O, y) = a^c 2 + bxy + c?/, write /(?/, x) , /(a, x) , and f(y, y). 

4. Define the differential coefficient of a function of x with regard to x. 
State what is the interpretation of the differential coefficient being positive 
or negative. 

431 



432 DIFFERENTIAL CALCULUS. 



5. Give a geometrical interpretation of -2 when x and y are connected 

dx 
by the relation /(x, y)=Q or y = <f>(x). 

6. Show that the derivative of a function with respect to the variable 
measures the rate of increase of the function as compared with the rate of 
increase of the variable. 

7. Tind geometrically the differential coefficients of cos x and sin x. 

8. Deduce from first principles the first derivatives of x n , sin x, tan x, 
tan _1 x, log a x, a x , a l °s x , log sin-- 

9. Find the derivatives of - and uv, with respect to x, where u and v 
are functions of x. 

10. Investigate a method of finding the derivative with respect to x of a 
function of the form {/(x)}<M x >, and apply it to differentiate x^ 1+x \ 

11. Differentiate — — — , lo g( cosa e^cos'mx, xe™ sx . \ og h + acosx 

(l + x*y x ' a + fccosx 



tan^e*, x m e ax sin w x, 



( 2 x sin log x \ 
[ x 2 - 1 ) 



12. Show that (1) Z> sin-i J^-=-^ = Z) CO s-iJ^^; (2) Z>sin-i^_^ 

+ 2 , sin -iVMES(IZZ) == o. 

a + 6x 

13. If x 2 ?/ 3 + cos x — sin x tan y — sin y = 0, show that 

dy _(—2xy s + sin x) cos 2 y + cos x sin ?/ cos ?/ 
ax — 3 x 2 ?/ 2 cos 2 «/ — sin x — cos 3 y 

14. Differentiate: (1) ^^^ + log VT^ 2 ; (2) tan-i Vb * ~ a * sin * 



-y/l _ x 2 a + o cos x 



(3) cos 



.iHacosx. ^ sin -i 6 + a sin x . ^ tan -i ^« 2 - &2 sin 



a + b cos x a + b sin x 6 + a cos x 



(6) Vmsin 2 x + ncos 2 x; (7) (2a* + x*)"^a* + x* ; (8) ^"^ 
(9) (cosx) sin *; (10) tan- 1 Vl + x + ^ 



(cosmx) n 



VI + * a - vT-^ 



[Answers to Ex. 14: (1) s[n ~ lx ; (2) ^ ~ a * ; (3) ^ZEE_ ; 

L q_ 2 \f 6 + acosx a + 6 cos x 

(4) vw=w . 6 VW=T> (6 1 (m _ B) H.i» 

a + &sinx a + fccosx 2 Vm sin 2 x + w cos 2 x 

> 7 v 4Va + 3Vx . ,ox mw (sin mx)™- 1 cos (mx - nx) . , q , rpn< ,^ sin x-: 

(7) -777=7' (8) (cosmx)«+i ' (9) (C ° SX) 

(cos 2 x log cos x — sin 2 x) ; (10) . x . •! 

VT^x* J * 



QUESTIONS AND EXERCISES. 433 



CHAPTER V. 

1. If the equation of a plane curve be y = 0(x), find the equations of the 
tangent and the normal at any point, and find the lengths of the tangent, 
normal, subtangent, and subnormal. 

2. Deduce the equation of the tangent at the point (x, y) on the curve 
y = /(x), when the curve is given by the equations x = 0(f), y = \p(t). 

X 

Prove that - + ^ = 1 touches y = be a at the point where the latter crosses 

u.i • a b 

the y-axis. 

3. Find an equation for the normal at any point on the curve whose 
equation is /(x. y) = 0. 

4. At what angle do the hyperbolas x' 2 — y 2 = a 2 and xy = b intersect ? 
Draw sets of these curves, assigning various values to a and b. 

5. Find the angle of intersection between the parabolas y 2 = 4 ax and 
x 2 = 4 ay. 

6. Find an expression for the angle between the tangent at any point of 
a curve and the radius vector to that point. Show that in the cardioid 

r = a (I + cos 0) this angle is — -\ — 

7. Determine the lengths of the tangent, normal, subtangent, and sub- 
normal, respectively, at any point of each of the following curves : (1) the 

I X 

hyperbola b 2 x 2 - a 2 y 2 = a 2 b 2 ; (2) the catenary y = J? (e« + c~«) ; (3) the 

parabola y 2 = 9 x. [Ans. (1) — V(a 2 - * 2 )(a 4 - e 2 x 2 ), — Va* - e 2 x 2 , 

ax a 2 

*~ a \ **5j (2) y 2 , £, °y , y Vy^^-, (3) 10, ?i, s, ^.] 

x a 2 Vy 2 - a 2 a Vy 2 - a 2 a 

8. Show that all the points of the curve y 2 = 4 a( x + a sin - ] at which 

V a) 

the tangent is parallel to the axis of x lie on a certain parabola. 

a 

9. (1) In the curve r=a sin 3 -, show that = 4^. (2) In the leinnis- 

o 

cate r 2 = a 2 sin 2 6, show that ^ = 2 0, = 30, subtangent = a tan 2 Vsin 2 0. 

10. Solve the following equations : (i) 4 x 3 + 48 x 2 + 165 x + 175 = ; 
(ii) 9 x 4 + 6 x 3 - 92 x 2 + 104 x - 32 = ; (iii) 16 x 5 + 104 x 4 + 73 x 3 - 277 x 2 
- 161 x + 245 = 0. 

11. Show that the condition that ax 3 + 3 6x 2 + 3 ex + d — may have 
two roots equal is (be — ad) 2 = 4 (ac — b 2 )(bd — c 2 ). 



484 DIFFERENTIAL CALCULUS. 

12. Prove, geometrically or otherwise, that provided /(x) satisfies a 
certain condition which is to be stated 

f(x + h) -f(x) =hf'(x + 8h), 
where d is a proper fraction. Show that it is possible that in this relation 6 
may have more values than one. 

13. If A is the area between the graph of /(x), the x-axis, a fixed ordi- 
nate, and the variable ordinate f(x) , show that — = f(x) . 

CHAPTER VI. 

1. Find the nth derivative of the product of two functions of x in terms 
of the derivatives of the separate functions. 

2. Find the fourth derivative of x 5 cos 3 x and the nth derivatives of 

1 r 3 

(i) x s cos ax; (ii) x 4 cos 4 x; (iii) tan -1 -; (iv) sin 3 x cos 2 x ; (v) — - — 



(vi) e ax sin bx. x x 2 1 

3. Show that 

(iii) WJ— g^ = 2 (~ 1)nn ! ; (iv) i) 3 (e sin *) = -e^*cosxsinx(sinx + 3). 
y J \l+x) (l+x)»+ 1 ' v J v ; v -r J 

4. If x = a(l - cos 0, V = a(nt + sin t), then ^M = - n cos t + 1 . 

dx 2 a sin 3 £ 

5. Derive the following : (i) If e»+xy-e=0, D 2 y = y ■ ( 2 - V) ey + 2x ^ 

x (e* + x) 3 

(ii) If x 4 + y* + 4 a 2 x?/ = 0, (y 3 + a 2 x) 3 ^ = 2 a 2 xy(x 2 y 2 + 3 a 4 ). (iii) If 

f?X 2 

dx 2 (Ax + by +f) 2 

6. Prove the following: (i) If y = sin (to tan -1 x), (1 + x 2 ) 2 — ^ + 

dx' 2 

2 x(l + x 2 ) ^ + to 2 ?/ = 0. (ii) If y = (x + Vx^^T)", (x 2 - 1)^ + x^ - 
w 2 2/=0. (iii) If*/ 2 =sec2x, y-\-*J=?> y &. (iv) If y=(l+a; 2 y*sin (mtan-ix), 

(1 + X 'i) ^M. _ 2(m - l)x^ + to (to - l)y = 0. 

dx 2 dx 

7. If aey + be~v + ce x — e - * = 0, determine a relation connecting the first, 
second, and third derivatives of y. 

CHAPTER VII. 

1. Write a note on the turning values of functions of one variable. 

2. Assuming/(x) and its derivatives to be continuous functions, investigate 
the conditions that /(a) should be a maximum or a minimum value of /(x). 



QUESTIONS AND EXERCISES. 435 

3. Show how you would proceed to find the maximum and minimum 
values of a single variable, and to discriminate between them. 

4. If f(x) have a maximum or minimum value when x = a, and f(x) be 
continuous at x — a, prove that f'(x) must vanish when x = a. Show by- 
means of a diagram that the converse is not necessarily true. Examine the 
case in which /(x) has a maximum or minimum value when x = a, and /'(a;) 
is discontinuous when x = a. 

5. If x 3 + 3 x-y + 4 y B = 1, show that \/| is the maximum and that | is 
the minimum value of y, where x can have all possible values. 

6. ABCD is a rectangular ploughed field. A person wishes to go from 
A to C in the shortest possible time. He may walk across the field, or take 
the path along ABC ; but his rate of walking on the path is double his rate of 
walking on the field. Show that he should make through the field for a point 

on BC distant b ^ from C, a and b being the leugth'of AB and BC 

respectively. v 3 

7. Prove that the greatest distance of the tangent to the cardioid 
r = a(l + cos 6) from the middle point of its axis is aV2. 

8. AB is a fixed diameter of a circle of radius a and PQ is a chord per- 
pendicular to AB ; find the maximum value of the difference between the two 
triangles APQ, BPQ for different positions of the chord PQ. 

9. Show that the point on the curve 4 ay = x 2 , which is nearest the point 
(a, 2 a), is the point (2 a, a). 

10. Show that the minimum value at which a normal chord of the ellipse 



ab 



— + y~ = 1 recuts the curve is tan -1 

a 2 6 2 a 2 - o 2 

11. Prove that the greatest value of the area of the triangle subtended at 
the centre of a circle by a chord, is half the square on the radius of the circle. 

12. A slip noose in a rope is thrown around a square post and the rope is 
drawn tight by a person standing directly before the vertical middle line of 
one side of the post. Show that the rope leaves the post at the angle 30°. 

13. Show that the maximum and minimum values of integral algebraic 
functions occur alternately. 

14. (i) Show that the points of inflexion on a cubical parabola y 2 = 
(x — a) 2 (x — b) lie on a line Sx + a = 4 b. (ii) Show that the curve 
y(x 2 + a 2 ) = a 2 (a — x) has three points of inflexion on a straight line. 
(iii) Show that the curve x 3 — axy + 5 3 = has a minimum ordinate at 

x = — - , and a point of inflexion at (— &, 0). 
V^2 



436 DIFFERENTIAL CALCULUS. 

15. Find where the following curves have maximum or minimum ordi- 
nates and points of inflexion respectively : (i) y = x i — 4 x 3 — 2 x 2 + 12 x + 4 ; 

(ii) y — xe x ; (iii) y = xe~ x ; (iv) y = xe~ x '. Ans. (i) x = — 1, 1, 3, 

1 ±|V3; (ii) x = - 2 j (iii) sc = 1, a; = 2 ; (iv) a =± — , a = 0, x =± Vj.l 

V2 J 

16. Find the inflexional tangent of the curve y = x — x 2 + x 3 . [A?is. 27 ?/ 
= 18x + 1.] 

17. Show that : (i) The cone of maximum volume for a given slant side 
has its semi- vertical angle = tan -1 V2; (ii) The cone of maximum volume 
for a given total surface has its semi-vertical angle = sin -1 i. 

18. Show the march of each of the following functions : (i)~ sin 2 x cosx ; 
(ii) sin 2 a; — x; (iii) x(a + x) 2 (a — x) 3 . 

19. Examine the following functions for maxima and minima : 

rn x(x 2 -l) . m a? + 2 s + 11 . (m 1-x + a* . ([ \ 1 + x + x 2 , 

^ > X i _ X 2 + i > y J X 2 + 4 x + io ' v J l + x - x 2 ' v J 1 - X + X 2 ' 

(v; x Vax - x 2 ; (vi) (x - l) 4 (x + 2) 3 ; (vii) (1 + x) 2 - (x - x 2 ) ; 

(viii) secx — x; (ix) sin x(l + cos x) ; (x) asinx + 6cosx; (xi) x x ; 

(xii) — - — Ans. (i) Two max., each = \ ; two min., each =— \ ; 

log x L 

(ii) max. = 2, min. = \ ; (iii) min. = | ; (iv) max. = 3, min. = \ ; (v) min. 
_ 3V3 fl2 . ^ min> _ q^ max> _ 12 4 . 93 + 77 . ( V ii) max< _ 0, m in. = 8 ; 

(viii) sin x = : ^-; (ix) max. = 1.299; (x) max. = Va 2 + & 2 , min. = 

A 

— y/a 2 + b 2 ; (xi) min. for x = -; (xii) min. = e. 

e J 

CHAPTERS VIII., IX. 

1. What is meant by partial differentiation ? 

2. State precisely the restrictions as to the function /(x, y) so that the 

d 2 / = ay 

dx cty d^ dx 



fj2-/* J32/* 

theorem ° * = "■' may hold, and prove the theorem 



Show that if f(x, y) = xi/- ^, the theorem does not hold for x=0, y=0, 

and explain why. x + V 

3. Explain the meaning of a partial derivative. In what sense may we 
logically speak of the partial derivative of c with respect to «, when c is a 
function of a and &, and a and b are both functions of x ? 

4. Prove Euler's theorem for a homogeneous function of x, y, z : 

x to. + yto + z d* = n<t> . 
dx dy d* 



QUESTIONS AND EXERCISES. 437 

5. If w be a homogeneous function of the nth degree in any number of 

variables x, y, z, ••-, then x^ + y— + z^- + ••• = nu. 
dx dy dz 

6. Verify that JL ( $*) = JL ( $*\ i n the case of each of the following 

dx\dy) dy\dx) 

functions: sin (x 2 y) , cos ( 2 ^ y X W x2 + y \ <f>(^ 

7. Verify the following : (i) If u = sin" 1 - + tan" 1 £, x^+w^=0. 

y x dx dy 

(ii) If v=(4a6-c 2 rt— = — (iii) If g=x a tan-i^ -y 2 tan-ig, d 2g 
o o y ' dc 2 da<3& V ^ x " y 5«5y 

= ^4 GO ^ V =f(V + «D +0 (V - ax), in general 2* = a 2 £* 

x- + y z Qx 2 dy 2 

(v) If u = log ^^ + 2 tan- 1 ?, dw = -i®l (^ ^_ x ^). ( v i) If w=tan-i t 
x+y y x*-?/ 4V * *' x 

0Tifu =^fjS + g + fH°- ( vii ) n« = rfnOn + w + ^) l 

r^V+r 1 -^ r ^ + 2(x + , + *) W = o. (viii) if w =^t?, 

dx-dy dz 1 — u 2 dx dy dz 

dx 2 ^ y dxdy dy 2 4 

8. Verify the following: (i) If (3a^L + 2\ ( ^ = ( a & + lW & 

\ dx J \dx 2 / \ dx J dx dx B 

f*z\* = /dz \*z m (ii) lH1 + y , ) (d^_ 2 \Jdyy dy^y 

\dy 2 ) \dy Jdy* K J y T y J [dx* J ] \dx) K U) dxdx 2 

znd y = z 2 + 2z,(z + l)^ = ( te^ + z 2 + 2z. (iii) K^ + -^_^ 
y y J dx* dxdx 2 K J dx 2 1 + x 2 dx 

+ v - = and x = tan z, ^ + y = 0. (iv) If (a + 6x) 2 ^ 

^(1+x 2 ) 2 ' dz 2 u * dx * 

+ A(a + bx)^- + By = F(x) and a + &x = c«, b 2 ^-+ b (A - b)^- + By 

dx dt 2 dt 

- f(?—-^\ • (v) If ^ - sec cosec 6^- + */n 2 tan 2 6 = and x = log sec 0, 
, \ b J dd 2 dd 

g+»*»=o. 

dx 2 

CHAPTER X. 

1. Define curvature of a curve. Find an expression for the radius of 
curvature of a curve whose equation is in the form y = f(x). 

2. Show that the curvature at any point of the curve given by x = <p(£) , 
y = xf/(t) is l^l — ~ - ^ , where accents denote differentiations with respect 
to t. O' 2 + V' 2 )* 

T 

3. For any curve /(r, 0) =0 show that radius of curvature = 



in which \p = tan -1 

dr 



rde] ' **H 1+ %) 



438 DIFFERENTIAL CALCULUS. 

4. Find the coordinates of the point on the parabola x 2 = 4 ay for which 
the radius of curvature is equal to the latus rectum. 

5. Show that at a point of undulation the tangent has contact of at least 
the third order. 

6. Show that the circle (4 x -8 a) 2 + (4 y - 3 a) 2 = 8 a 2 and the parabola 
\/x-\-Vy=\ / a have contact of the third order at the point (-, -Y Find 

the order of contact of the curves y = x z and y =3x 2 — 3 x + 1. 

7. Show that the circles of curvature of the parabola y 2 = 4 ax for the ends 
of the latus rectum have for their equations x 2 + y 2 — 10 ax ± 4 ay — 3 a 2 = 0, 
and that they cut the curve again in the points (9 a, =f 6 a). 

8. Find the radius of curvature of each of the following curves : 
(i) The card ioid r* = a?coa%6. (ii) y = 2 x + 3x 2 - 2 xy + y 2 at (0,0). 
(iii) xy 2 = a 2 {a + x) at (— «, 0). (iv) The tractrix a; = a log cot — a cos (9, 
y = a sin 0. (v) y = x — sin x at the origin, and where x = - • (vi) The expo- 

nential curve y = ae c . (vii) r m = a m cos md. (viii) r = asm nd at (0, 0). 
(ix)r 3 =a 3 cos3 0. \ Ans. (i) fVar. (ii) |V5. (iii) | a. (iv)-acot0. 

(v)0,2V2. (vi) fr a + y 2 )* . (vii) ^ (viii)ina. (ix) -*Ll 



CHAPTER XIII. 

1. Define an asymptote to a curve. Derive a method of finding the 
asymptotes of an algebraic curve whose equation in Cartesian coordinates is 
of the nth degree. 

2. Show that the asymptotes of the cubic x 2 y — xy 2 + y 2 + xy-\-x — y = 
cut the curve again in three points which lie on the line x + y = 0. 

3. Find the asymptotes of the curve xy 2 — x s -{- 2 x 2 -\- 3 y -\- x — 1 = 0. 
Show that the points at a finite distance from the origin in which the 
asymptotes cut the curve lie on the line 3y-{-2x — 1=0. 

4. Draw the curve x 2 y = x z — a 3 . Show that it has an asymptote which 
crosses the x-axis at an angle tan -1 3. 

5. Find the asymptotes of the following curves: (\)xy 2 — x 2 y=a 2 (x+y)-\-b*. 

(ii) 1 + y = e x . (iii) x 3 — xy 2 + ay 2 — a 2 y = 0. (iv) (x 2 + y 2 ) (y 2 — 4 x 2 ) 
+ 4y 2 (x-l) + x 2 (4x + 3) = 0. (v) (X - 2 a)y 2 = x* — a 3 . (vi)x 3 + 3?/ 3 
=a 2 (y-x). (vii) x 3 +2 z 2 y-\-xy 2 —x 2 -xy+2 = 0. (viii) r sin 2 d = a cos 3 0. 
(ix) y* = x 2 (2a-x). 

6. Find the asymptotes of the curve x s y — xy z + 6 a 2 xy + a 2 y — 16 a 2 x = 0. 
Show that the origin is a point of inflexion. 



QUESTIONS AND EXERCISES. 439 

7. Define a family of (plane) cuiwes, and the variable parameter of the 
family. Define the envelope of a family of curves. Define an ultimate 
intersection of a family of curves. Define the locus of the ultimate intersec- 
tions of a family of curves. Illustrate the definitions by concrete examples 
and diagrams, and furnish any explanations you may think necessary. 

8. Show that in general the locus of ultimate intersections of the family 
touches each member of the family. Show that this locus is, in general, the 
envelope of the family. Explain the necessity of the qualifying phrase "in 
general." 

9. Explain the method of finding the envelopes of the curves /(a:, y, t)=0, 
where t is a variable parameter. 

10. "Write a note on "singular points of curves," explaining what they 
are, giving illustrations, and showing how to find them. 

11. Ellipses of equal area are described with their axes along fixed straight 
lines. Show that the envelope consists of two equilateral hyperbolas. 

12. Prove that the circles which pass through the origin and have their 
centres on the equilateral hyperbola x 2 — y 2 = a 2 envelop the lemniscate 
(x 2 + y 2 ) 2 = 4:a 2 (x 2 -y 2 ). 

13. P is a point on a parabola of which A is the vertex. Find the equa- 
tion of the curve touched by all circles described on AP as diameter. 

14. A circle passes through the origin, and its centre lies on the parabola 
y 2 = 4 ax. Show that the envelope of all such circles is a cissoid. 

15. A straight line moves so that the product of the perpendiculars on it 
from two fixed points (± c, 0) is constant (= k 2 ). Show that its envelope is 

the ellipse — \- %- = 1, or the hyperbola ■ — ^- = 1. 

* k 2 + c 2 k 2 Jr c 2 -k 2 k 2 

16. Eind the envelope of circles passing through the centre of an ellipse 
a 2 y 2 + b 2 x 2 = a 2 b 2 and having centres on the circumference of the ellipse. 
[Ans. {x 2 + y 2 ) 2 = i(a 2 x 2 + b 2 y 2 ).~] 

17. Ellipses are described having their axes coincident in direction with 
those of a given ellipse, and lengths of axes proportional to the coordinates of 
a variable point on the given ellipse. Show that the ellipses all touch four 
straight lines. 

18. Eind the equation of the envelope of the line £sin# + ycosa = 
a sin a cos a. 

19. From a fixed point on the circumference of a circle chords are drawn, 
and on these as diameters circles are drawn. Show that the envelope of the 
series of circles is a cardioid. 

20. If a cannon is fired at an elevation 0, and the projectile has an initial 
velocity equal to that attained by a body in falling h feet, the equation of the 
parabolic path, referred to horizontal and vertical axes through the point of 



440 DIFFERENTIAL CALCULUS. 

a* 2 
projection, is y = x tan 9 — — sec 2 0. Find the envelope of the paths for 

different elevations. 

CHAPTEES XV., XVI. 
1. A function f(x) is denned by an infinite series f(x) = ^ 0»(*) 5 st ate 

n=l 
w=oo 

and prove a sufficient condition that the equation — f(x) = X — <t> n (%) may 

1 . dx ^ dx 
be true. »=i 

2. Write a note on the conditions under which (1) the integral, (2) the 
differential coefficient of an infinite series, may be obtained by integrating or 
differentiating the series term by term. 

3. Prove that if f(x) be a continuous function of x, then 

f(x + h) = f{x) + hf'{x + 6K), 
where < < 1. 

Show clearly how this proposition may be applied to prove Taylor's theo- 
rem, and specify the circumstances in which the theorem as you state it is true. 

4. Prove Taylor's theorem for the expansion of f(x + h) in ascending 
powers of h, carefully specifying the conditions which f(x) must satisfy. 
Find an expression for the remainder after n terms of the series have been 
written down. 

5. State Maclaurin's theorem, and give the conditions under which it is 
applicable to the expansion of functions. Derive the theorem. 

6. Expand in series of ascending powers of x the functions : (i) cos mx. 
(ii) tan-^a + x). (iii)_ sin (m sin" 1 x). (iv) (1 + y) x , where y < 1. 
(v) e mx + e~ mx . (vi) e Vx + h , 4 terms. 

7. Expand the following functions in powers of x : (i) e sin x . (ii) tan -1 x. 
(iii) cot" 1 x. \Ans. (i) l + x + ix 2 -ix 4 - T ^ 5 + —. (ii) For 
values of x from x = — 1 to x = 1, x — | x 3 + £ x 5 — }x 7 + ••• ; for [ x | > 1, 

£_I + J_ i + .... (iii) For |x|<l, £-z + is3_i x 5 + .... for 

2 x Sx 6 ox b 2 

l*l> 1 'i-8P + 6S--] 

8. Calculate the values of the following : 

(i) J *x J Vl — x 2 dx. (ii) \ x xcotxdx. (iii) ( e* 2 dx. (iv) jVsinxcfo. 
(v) J o * ^ dx. [.to. (i) f aj*{l - i *■ - A «* - A * 6 + -)■ 

en) X -*L*L**.„. (m) 2 (i+i+—i—+ 1 + i +...V 

v J 9 225 6615 v J \ 3 1-2-5 1-2.3-7 1-2.3-4-9 / 

^ ; 2! i "3! + 4! 6! 7! 8 ! + '"' KJ 3.316.5! ""J 



QUESTIONS AND EXERCISES. 441 

CHAPTERS XVIII.-XXII. 

1. Explain and illustrate the meaning of integration. 

2. If f(x) be finite and continuous for all values of x betwe en a and b, 
prove that lini^/i {/(«) + f(a + h) + f(a + 2 A) + — + /(a + n- 1 ft)} is 

0(6) - 0(a), where A = ^-^ and — 0(x) = f(x). 
n dx 

3. Explain fully how it is that the area included between a curve, the 
axis of x, and two ordinates corresponding to the values Xo and Xi of x is 

represented by the definite integral I 1 ydx. 

4. Give an outline of the reasoning by which it is shown that the area 
bounded by the two curves y — 0(x) and y = ^(x), and the two ordinates 

x = aandx=6, is i {4>(x)— \f/(x)}dx. 

5. Prove Simpson's or Poncelet's rule for measuring a rectangular field, 
one of whose sides is replaced by a curved line. 

The graph of y = x 2 is traced on a diagram. If be the point (0, 0) on 
it, Pthe point (10, 100), and PJf the ordinate from P, find the area of OMP 
cut off between 031, MP, and the curve, by taking all the ordinates corre- 
sponding to integral values of the abscissas, and applying the rule you adopt. 
Tell exactly by how much your calculation is wrong. 

6. Show how to find the volume of the surface generated by the revolu- 
tion of a given curve about an axis in its plane. 

7. Find the area cut off between the parabola y = x 2 and the circle 
x 2 + t = 2. 

8. Trace the curve whose equation is a 4 ?/ 2 = x 4 (a 2 — x 2 ), and find the 
whole area enclosed by it. 

9. Show that the area included between the curve y 2 (2 a — x) = x 3 and 
its asymptote is 3 ira 2 . 

10. Determine the amount of area cut off from the circle whose equation 
is x 2 + y' 2 = 5 by a branch of the hyperbola whose equation is xy = 2. 

11. Trace the curve ay +2 x(x — a) = 0. Find the area of the closed por- 
tion contained between the curve and the axis of x. If this portion revolves 
round the axis of x, find the volume generated. 

12. A curved quadrilateral figure is formed by the three parabolas 
y 2 - 9 ax + 81 a 2 = 0, y 2 - 4 ax + 16 a 2 = 0, y 2 - ax + a 2 = 0, the other boun- 
dary being the axis of x. Find the area of the quadrilateral. 

13. Show that the volume of the solid generated by revolving about the 
x-axis, an arc of a parabola extending from the vertex to any point on the 
curve, is one-half the volume of the circumscribing cylinder. 



442 DIFFERENTIAL CALCULUS. 

14. Determine the curve for any point of which the subtangent is twice 
the abscissa and which passes through the point (8, 4). 

15. Write the equation including all curves that have a constant sub- 
normal. Determine the curve which has a constant subnormal and which 
passes through the points (0, h), (6, Jc), and find what is the length of its 

constant subnormal. [Aus. by 2 = (k 2 - h 2 )x + bh 2 ; fc2 ~ h2 .l 

16. In what curve is the slope at any point inversely proportional to the 
square of the length of the abscissa ? Determine the curve which has this 
property and passes through (2, 5), (3, 1). 

17. State and derive the rule known as "integration by parts. 1 ' Apply 
it to find j x* log x dx. 

18. Show that if the integral of /(x) is known, the integral of / -1 (x), the 
function inverse to /(as) , can be found. 

f(x\ 

19. Show how to integrate I=+\^-, where fCx) and 00*0 are rational 

00*0 
integral functions of as, and give some of the standard types for the integrals 
on which the value of I may be made to depend. Show how to integrate the 
fraction when the equation 0(x) = has repeated imaginary roots. 

20. Show that if fCu, v) is a rational function of u and v, f x, \l ax + b \dx 

ax + b ^ Vcx + d) 

can be rationalised by means of the substitution "*" = z n . 

ex + d 

21. What is meant by a formula of reduction for an integral ? 

Investigate formulas of reduction for the following : (i) \ sin™ 8 dd 

f c x m 

hi which m is an integer ; (ii) \ sin m 8 cos n 8 dd ; (iii) \ ■ , dx ; 

„ J J Vfl 2 + x 2 

(iv) I x n sin x dx. 



5. Explain how it is that y cos 2n + 1 8 dd = 0. 

dx 

(x — p) Vax 2 + 2 bx + c 



r dx 

23. Evaluate I . by means of the substitution 

J (x—p) Vax 2 + 2 bx + c 

y(x — p) = Vax 2 + 2 bx + c. 

24. Evaluate the following integrals, and verify the results by differentia- 
tion 



J (l + a; 2)i J o X « + * J f sin^cos3^ J f cos |^ 

f d0 r x 1 dx C dx C dx 

J a 2 cos 2 8 + b 2 sin 2 0' J x 12 - l' Jx(3 + 4x 5 ) 3 ' J 3 sin x + sin 2 as' 

(x*(a + x) sete, f 2x + 1 — ^ C x * tan -i ^ ^ T e 2x S i n 2 a. cte> 

J ./ x 2 — 4x4-3 J J 



QUESTIONS AND EXERCISES. 443 






(fa 



dx . r(x±l)dx 



ax , /• 

x V— x 2 + 5x - 6 J 



a; v — x 2 + 5 35 — 6 J Vx 2 4- x + 1 



CHAPTERS XXIV., XXV. 

1. Find an expression for the area bounded by a curve given in polar 
coordinates and two straight lines drawn from the pole. 

2. Show how to find the length of the arc of a plane curve whose equa- 
tion is given (i) in rectangular Cartesian coordinates, (ii) in oblique Carte- 
sian coordinates, (iii) in polar coordinates. 

3. Investigate a formula for finding the superficial area of a surface of 
revolution about the axis of x. 

4. Trace the curve r 2 = a' 2 cos 3 0, and find the area of one of its loops. 

5. Show that in the logarithmic spiral, r = a , the length of any arc is 
proportional to the difference between the vectors of its extremities. 



6. Find the area of the curve r v a 2 + b 2 = (a 2 + &' 2 ) cos 6 + a 2 . 

7. Find* the surface of a spherical cup of height h, the radius of the 
sphere being B. 

8. Find the average value of sin x sin (a — x) between the values and 
a of the variable x. 

9. Find the volume bounded by the surface '\/- + \/- + \/-=-l an( * tne 
coordinate planes. a c 

10. The axis of a cone is the diameter of a sphere through its vertex ; 
find, in terms of its vertical angle, the volume included between the sphere 
and the cone, and examine for what angle it is greatest. 

11. Determine the areas of each of the following figures : (i) The segment 
cut off from the parabola y' 2 = 4 ax by the line 2x — 3y + 4a = 0. (ii) The 



2. 2 

V + fy\ 3 _ 1 (iii) T j ie e volute of the ellipse (ax)% -f- (by)* 



(a 2 - b 2 )K (iv) The figure bounded by the ellipse 16 x 2 + 25y 2 = 400, the 
lines x = 2, x = 4, and 2 y + x = 8. (v) The curve (x 2 + y 2 ) 2 = a 2 x 2 + b 2 y 2 . 
(vi) The oval y — x 2 + V(x — 1)(2 — x). (vii) The loops of the curve 
a 2 y 2 = x 2 (a 2 — x 2 ). (viii) The segment of the circle x 2 + y 2 = 25 cut off by 
the line x + y = 7. (ix) The area common to the ellipses & 2 x 2 + a 2 y 2 = a 2 b 2 , 

a 2 x 2 + b 2 y 2 = a 2 b 2 . [Ans. (i) 1 a 2 . (ii) | rrab. (iii) f ir ( a * ~ &2)2 . 

( V ) 7r ( a2 + 62 ) . (vi) |. (vii) Each fa 2 . (viii) ^_ sin -i ^ _ |. 

-1* 



(ix) 4 a6 tan-ill 
a J 



444 DIFFERENTIAL CALCULUS. 

12. Find the volume and the area of the surface generated by the revolu- 
tion of the cardioid r = a(l — cos 6) about the initial line. [Area = - 3 ^ ?m 2 .] 

13. Show that the volume enclosed by two right circular cylinders of 
equal radius a whose axes intersect at right angles is - 1 / a 3 , and the surface 
of one intercepted by the other is 8 a 2 . 

14. Show that the volume included between the surfaces generated by 
the revolution of a hyperbola and its asymptotes about the transverse axis 
and two planes cutting this axis at right angles is the same, no matter where 
the sections are made, provided that the distance between the planes is kept 
constant. 

15. The parabola y 2 — 6 x intersects the circle x 2 + y 2 = 16. Show that 
if the larger area intercepted between the curves revolves about the x-axis, 
the volume generated is 60 ir cubic units ; and show that if the smaller area 
intercepted revolves about the y-axis the volume generated is £|± V3 w cubic 
units. 

16. An arc of a circle of radius a revolves about its chord. Show that if 
the length of the chord is 2 act, volume of the solid = 2 7ra 3 (sin a — | sin 3 a 
— a cos a), surface of the solid = 4 7ra 2 (sin a — a cos a). 

17. Tund the area of the segment cut off from the semi-cubical parabola 
27 ay 2 = 4 (x — 2 a) 3 by the line x = 5 a. Also find the volume and the area 
of the surface generated by the revolution of this segment about the x-axis. 

, f 7V2 



tftf, 7ra2(^p + f log( V2 + l)}.] 



18. A number n is divided at random into two parts. Show that the 
mean value of the sum of their squares is f n 2 . 

19. Show that the mean of the squares on the diameters of an ellipse, that 
are drawn at points on the curve whose eccentric angles differ successively 
by equal amounts, is equal to one-half the sum of the squares on the major 
and minor axes. 

20. Prove that the mean distance of the points of a spherical surface of 

a 2 c 2 

radius a from a point P at a distance c from the centre is c + — or a -f — , 

according as P is external or internal. 

CHAPTER XXVII. 

1. Solve the following equations : 
(1) x 2 y dx - (x 3 + y*)dy = 0. (2) 3 e* tan y dx + (1 - e 35 ) sec 2 y dy = . 

(3) (x 2 -4xy-2y 2 )dx + (y 2 -4xy-2x 2 )dy = 0. (4) xDy-y = x\/x 2 +y 2 . 
(5) (x 2 + y 2 ) (xdx + y dy) =a 2 (xdy-y dx) . (6) (x 2 + \)Dy + 2 xy = 4 x\ 
(7) 6(x + l)Dy = y-y x . (8) i> 3 - 4 xyp + 8 y 2 = 0, in which p = D x y. 

(9)^M = *V. (10)^+5-^2/=!. (ll)y=x 2 -\p 2 . (I2)x+2py=p 2 x. 
dx x dx x 



QUESTIONS AND EXERCISES. 445 

(13) D<>y + 2 Dfy + D,y = 0. (14) g_ 3 g + 4|_2y = 0. 

-2 + f »Hfr CI*) 2*^ = 1. 

(20) y H + § (i - 2 y ) = °- (21) 2 ^ ^ + a " = (Z>W - 

[/SbZutfoni; : (1) 3 y 3 log ?/ = x 3 + c. (2) tan y — c(l — e 1 ) 3 . (3) x 3 — 6 x 2 y 
- 6 zt/ 2 + ?/ 3 = c. (4) 2 y = x(ce* - ce~ x ) . (5) x 2 + y 2 = 2 a 2 tan.- 1 ^ + c. 



(6) 3(x 2 + l)y = 4 x 3 + c. (7) Vx + 1(1 - y 3 ) = cy 3 . (8) y = c(x - c) 2 . 

(9) 2 ?/- 5 = ex 5 + 5 x 3 . (10) y = x 2 (l + ce x ). (11) (x 2 + ?/) 2 (x 2 -2y) 
+ 2 x(x 2 - 3 y)c = c 2 . (12) l+2cy = c 2 x 2 . (13) y = d + e-*(c 2 + c 3 x). 
(14) y = e z (ci + c 2 cos x + c 3 sin x). (15) y = c\ + c 2 x + e* (c 3 + c 4 x). 

(16) xy = Ci logx — log (x — 1) + c 2 . (17) y = x (ci + c 2 log x) + c 3 x _1 . 

(18) sin(c 1 -2V2y)=c 2 e- 2a: .' (19) x=- Vcy 2 -y+— — hycos" 1 (2 cy-T) +d. 

c 2cVc 5 

(20) 2 x = log(y 2 + ci) + c 2 . (21) 15 c x 2 y = 4(cix + a 2 Y + c 2 x + c 3 .] 

2. Find the singular solutions of : 
(1) x 2 p 2 -3 xyp + 2 y 2 +x 3 =0. (2) xp 2 -2 yp + ax=0. (3) Solve equation (2). 
^Solutions : (1) x 2 (y 2 - 4 x 3 ) = 0. (2) y 2 = ax 2 . (3) 2 y = ex 2 + -•] 

MISCELLANEOUS. 

1. How far does the symbol — obey the fundamental laws of algebra ? 

dx 

2. Prove that if D denote — , and f{D) be any rational algebraic func- 
tion of D, then f(D)uv = uf(D)v + Duf'(D)v + — f"(D)v + •••. 

3. If denote any function of x, prove that — k^ri = n ^ + x — *• 

dx n dx 11 - 1 dx n 

By this theorem or otherwise find the value of D b (x sin mx). 

4. H x = e*, prove that ±(±-l\(*.-*\.J±-n + l)u = *^ 

dd\dd j\d6 I \dd ) dx*' 

where u is any function of x. Prove also that ( — x — ) u = [ — ) x ( — ) u. 

\dx dx) \dx] \dx) 

5. If 0(x) is a function involving positive integral powers of x, prove the 
symbolic equation l~— ( e ax ■ u ]~| = e ax (pla + —\u. 

6. Show how to find the values of -^- and —4 when x and y are con- 

dx dx 

nected by the equation /(x, y) = 0. 



446 DIFFERENTIAL CALCULUS. 

7. If u = /(x, y) and if x = <j>(t), y = \f/(t), state and prove the rule for 
obtaining the total derivative of u with respect to t. 

Q2 U I d 2 u d 2 u\ 

If x = r cos 0, y = r sin 0, transform (x 2 — y' 2 ) - — — -f xy —^ — ^— } into 

dxdy \dz 2 6V/ 

an expression in which r and are the independent variables. 

8. Calculate the nth derivative of (sin -1 x)' 2 . Show by the use of Mac- 
laurin's theorem that (sin- 1 ^) 2 = 2 — + - — + ^~±±--\- ... \, 

K J \2 3.4 3.5.6 ; 

9. The curves u = 0, zt' = intersect at (x, ?/) at an angle a. Show that 

du du' du' du 
dx dy dx dy 



tan « 



dudu_. du' Qu 
dx dy dx dy 



X 2 ) ft X 2 V 2 

Show that the curves — \- • - = 1 and 1- %— — 1 intersect at right angles 

if a?-b* = a' 2 -b' 2 . a ° a ° 

10. Show that the total surface of a cylinder inscribed in a right circular 
cone cannot have a maximum value if the semi-angle of the cone exceeds 
tan" 1 1. i.e. 26° 31'. 

11. Through a diameter of the base of a right circular cone are drawn two 
planes cutting the cone in parabolas. Show that the volume included between 

these planes and the vertex is — of the volume of the cone. 

3tt 

12. Calculate the area common to the cardioid r = a (1 — cos 6) and the 
circle of radius | a whose centre is at the pole. 

13. Find the area and the perimeter of the smaller quadrilateral bounded 
by the circles x 2 + y 2 = 25, x 2 + y 2 — 144, and the parabolas, y 2 = 8 x, 

yl + 12 (X + 2) = 0. 

14. Given the cardioid r = 4 (1 — cos 9) and the circle of radius 6 whose 
centre is at the cusp, find the length of the circular arc inside the cardioid 
and the lengths of the arcs of the cardioid which are respectively outside the 
circle and inside the circle. 

15. If a curve be defined by the equations — — = — ^— = , find an ex- 

0(0 K0 /(0 

pression for the radius of curvature at a point whose parameter is t. 

16. Expand (by any method) x 3 cosec 3 x in a series of powers of x as far 
as the term in x 4 . At what place of decimals may error come in by stopping 
at this term, when x is less than a right angle ? 

17. Trace the curve x i + y* = a 2 xy, and find the points at which the tan- 
gent is parallel to an axis of coordinates. Find the area of the loop. 

18. Trace the curve x = a sin 2 6 (1 + cos 2 0), y = a cos 2 d (1 — cos 2 0). 
(a) Prove that is the angle which the tangent makes with the axis of x, and 
obtain the equation of the tangent to the curve. (&) Find the length of the 
radius of curvature in terms of 0. 



QUESTIONS AND EXERCISES. 447 

19. Find ^ under each of the following conditions : (i) x 3 = e tan V * 2 /. 

dx 

(ii) y = e x * tan -1 x. (iii) e x + x = ey + y. (iv) y = • (v) sin (a;?/) 

_ e xy _ X 2y = o. x + Vl-x 2 

20. Four circles x 2 + y 2 = 2 ax, x 2 + y 2 = 2 ay, x 2 + y 2 = 2 bx, x 2 + y 2 = 2 by, 
form by their intersections in the first quadrant a quadrilateral ; prove that 

the area of this is (a 2 + & 2 ) cot" 1 2 ab - - (a - b) 2 . 

a 2 — b 2 

21. Prove that the area of a sector of an ellipse of semi-axes a and b be- 
tween the major axis and a radius vector from the focus is — (0 — e sin <j>), 

where is the eccentric angle of the point to which the radius vector is 
drawn. 

22. Trace the curve xy 3 = a 4 ; and find whether the area between it, a 
given ordinate, and the coordinate axes is finite. 

Show also that if the tangent at P meet the axis of x in T, then M T = 3 OM , 
where M is the foot of the ordinate at P, and is the origin. 

23. If u be a homogeneous function of n dimensions in x and y, show that : 

dx 2 dxdy dy 2 dx 2 dxdy dx 

(iii) «^- + f i5 == (»_l)|» (iv) (x|- + y |-W = n 2 W . 

24. Prove the following : (i) If u = sin" 1 (xyz), dududU- tan 2 w sec u. 

dxdydz 

(ii) If w = log (tan x + tan ?/ + tan «), sin 2 x^ + sin 2 y^ + sin 2 s^ = 2. 

dx dy dz 

(iii) If u = log (x» + 2/3 + S 3 _ 3 jqp) i» + 5» + ^ = 3 • (iv) If 

dx 3y 6^ x -f y + 

tan 2 x tan 2 ?/ tan 2 s, <fa = ±u f 



Ix sin 2 ?/ sin2^ y 

25. If & be the radius of the middle section of a cask, a the radius of either 
end, and h its length, show that the volume of the cask is y 1 ^ ir (3 a 2 + 4 ab 
+ 8 b 2 )h, assuming that the generating curve is an arc of a parabola. 

26. 031 is the abscissa, MP the ordinate of a point P(xi, y{) on the 

x 2 w 2 
hyperbola - — — =1, (xi, ?/i, both being positive). If A is the vertex nearest 
a 2 b 2 

P, show that area AMP = \ X\y\ — \ ab log ( ^ + — ) , and area sector OAP 

27. Show that the mean of the squares on the diameters of an ellipse that 
are drawn at equal angular intervals is equal to the rectangle contained by 
the major and minor axes. 



448 DIFFERENTIAL CALCULUS. 

28. Find the mean square of the distance of a point within a square from 
the centre of the square. 

29. Through a diameter of one end of a right circular cylinder of altitude 
h and radius a two planes are passed touching the other end on opposite sides. 
Show that the volume included between the planes is (tt — ±)a 2 h. 

30. Show that the integration of the expression f(x, y)dxdy may be per- 
formed in any order, provided the limits of x and y are independent of each 
other. 

31. Evaluate ( ( ( x a yPzy dx dy dz taken throughout the space bounded 
by the coordinate planes and the plane x + y -f z = 1. 

32. Prove geometrically or otherwise that xdy — ydx=r 2 dd, and show that 
the area of a closed curve is represented by \ \ (xdy — y dx). 

33. The equation to a curve being written in terms of the polar coordi- 
nates r and 0, p being the perpendicular from the pole to the tangent and 

u = -, show that, - = w 2 -j- ( — V. 

r V \dd) 

34. If a is a first approximation to a root of the equation f(x) = 0, deter- 
mine graphically or otherwise the conditions under which a — ^ a ' is a valid 
second approximation. ^ ^ 

35. If f(x) be a finite and continuous function of x between x = a and 
x = 6, show that a value X\ of x, lying between a and 6, may be found such 
that/'O!) = {/(&) -f(a)} - (6 - a). 

If the function be x c + cx, find the point in question when a = a and b=2a, 

and thence show that in this case X\ is such that a ~ Xl is constant for all 
values of a. b ~ Xl 



36. Find the radius of curvature of the curves : (i) limacon r=a cos 6+b, 
iere r = -; (ii) ay 2 =(x—d)(x — b) 2 at (a, 0). Trace the curves. Arts. 



(i) 2aS ; (ii) fe=2!.l 

37. (1) Trace the curve r=a-\-b cos 0, #>&>0 ; find its area. (2) Find 
the area of the loop of y 2 = (x — 1) (x — 3) 2 . (3) Find the area between the 

x-axis and one arch of the harmonic curve y=b sin -• Ans. |(2 a 2 +6 2 )7r, 

38. Trace the curve 9 y 2 = (x + 7) (x + 4) 2 . Find the area and the length 
of the loop, and the volume and area of the surface generated by the revolu- 
tion of the loop about the x-axis. [Ans. |V3, 4V3, f tt, 3 7r.] 



QUESTIONS AND EXERCISES. 449 

39. Find the limiting values of: (i) log ** sm ° ' , when 0=ir : (ii) f*°S?\i 

w (V 2 -0 2 )0 \ x ) ' 

when £ = cc ; (iii) — x " ~ x , when x = 1 ; (iv) — - .when 

1-ce + logx 2x 2 2xtan?rx 

x = 0; (v) ( S -^V 2 , whenx = 0; (vi) ^-=-^ , when x = ; (vii) £££, 
when x = a. 

40. Find the mass of an elliptic plate of semi-axes a and 6, the density- 
varying directly as the distance from the centre and also as the distances from 
the principal axes. 

41. From a fixed point A on the circumference of a circle of radius a, the 

perpendicular AY is let fall on the tangent at P. Prove that the greatest 

3 V3 
area APY can have is — — a 2 . 
8 

42. A rectangular sheet of metal has four equal square portions removed 
at the corners, and the sides are then turned up so as to form an open rec- 
tangular box. Show that the box has a maximum volume when its depth is 
^(a + b — Va 2 — ab + & 2 ), a and b being the sides of the original rectangle. 

43. Two ships are sailing uniformly with velocities u, v, along straight lines 
inclined at an angle 8 : show that if a, b, be their distances at one time from the 
point of intersection of the courses, the least distance of the ships is equal to 

(av — bu) sin 6 

(w 2 + v 2 — 2uvcosd)% 

44. A right circular conical vessel 12 inches deep and 6 inches in diameter 
at the top is filled with water : calculate the diameter of a spherical ball which, 
on being put into the vessel, will expel the most water. 

45. A statue a feet high is on a pedestal whose top is b feet above the level 
of the observer's eyes. How far from the pedestal should the observer stand 
in order to get the best view of the statue ? \_Ans. V&(a + b) feet.] 

46. The lower corner of a leaf, whose width is a, is folded over so as just 
to reach the inner edge of the page : find the width of the part folded over 
when (1) the length of the crease is a minimum, (2) when the area of the tri- 



47. (1) Show that the cylinder of greatest volume for a given surface has 
its height equal to the diameter of the base, and its volume equal to .8165 of 
that of the sphere of equal surface. 

(2) Show that the cylinder of least surface for a given volume has its 
height equal to its diameter, and its surface equal to 1.1447 of that of the 
sphere of equal volume. 



450 DIFFERENTIAL CALCULUS. 

48. Trace the graph of y = sm2x ~ sm x . Find the angles at which it 

COS X 

crosses the z-axis, and show that its finite maximum distance from the z-axis 
is (2! - l)i 

49. An ellipse, whose centre is at the origin and whose principal axes coin- 
cide with the axes of x and y, touches the straight line qx-\-py=pq ; find the 
semi-axes when the area of the ellipse is a maximum, and also the coordinates 
of its point of contact with the given line. 

50. Find the volume of the greatest parcel of square cross-section which 
can be sent by parcel post, the Post-office regulations being that the length 
plus girth must not exceed 6 feet, while the length must not exceed 3 feet 
6 inches. 



INTEGRALS. 

FOR EXERCISE AND REVIEW. 

The following list of integrals provides useful exercises in 
formal differentiation and integration. It will also afford some 
assistance in the solution of practical problems as a table of refer- 
ence. Those who have to make considerable use of the calculus 
will find it a great advantage to have at hand Peirce's Short Table 
of Integrals* (Ginn & Co.). 

GENERAL FORMULAS OF INTEGRATION. 

Formulas A , J9, C, pages 294, 295 ; formula for integration by parts, 
page 298. 

FUNDAMENTAL ELEMENTARY INTEGRALS. 

Formulas I.-XXVI., pages 293, 294, 301, 302. (These should be mem- 
orised.) 

REDUCTION FORMULAS FOR (x ±m (a + bx n )^dx. 

[Here X denotes (a + 6x n ).] 

1. (x^XPdx = ,f -J^lX^l _ a(m - » + 1) ( x m-n XPdx , 
J o(np + m + l) b(np + m+l)J 

2 Cx*»XP dx = * m+1 X p+l _b(m + n + np + l) C x m + n XP dx . 
J a(m + 1) a(m + l) J 

3. f x™XP dx = octn+1XP + ™1P — f ^JP-1 dx. 

J m + np + 1 m + np + U 

4. (#*X* d X = - ^ + '^ +1 + » + » + np + 1 C xmXP+ l ax , 
J an(p + 1) an(p + 1) J 

* There are two editions, the briefer edition of 32 pages and the revised 
edition of 134 pages. 

451 



452 DIFFERENTIAL CALCULUS. 

5. (W* dx = *"- n+1 ^ +1 _ m-n + 1 C xm - nXP+ i dx . 
J bn(p + l) bn(p + V)J 

6. (W* <fa = XW+1XJ? - -^- f Z-+-JTP-1 (fa. 
J m + 1 m + w 

7 f_(fa_. 1 (w — n + np — 1)6 f <fa 

J «"»Ji» ~~ (m - l)a* m - 1 XP- 1 (m — l)a J x m ~ n Xp 

8 [• dx 1 . m — n + np — I f dx 

J x m XP ~ an(p — l)x m - 1 XP" 1 an(p — 1) J x m Xp~ 1 

9 f XPdx _ Xp +1 b<jm-n- np -\) C XPdx ^ 
J x m a(m — l)x m_1 a{m — 1) J x m ~ n 

10 CX p dx Xp anp f Xp^dx 

J x m (np — m + \)x m ~ 1 np — m + 1 J x m 

-- f x m dx _ x m ~ n+l a(m - n + 1) f x m ~ n dx r 

J Xp ~ b(m-np + 1)Xp~ 1 b(m - np + \)J Xp 



-o f x m dx _ x m+1 m + n — np + 1 f x m dx 

J Xp ~ an(p - 1)Xp-* an(p - 1) J Xp- 1 ' 

13. f *? = 1 f 5 + (2n -3)f ^ -T 

J (a + &X 2 )" 2(w - \)a |_(a + &X 2 )"- 1 v J (a + fcx 2 )"" 1 .] 

Put a 2 for a, 6 = 1, and compare with Ex. 3, Art. 118. 

.. f x 2 dx — x j 1__ _ f dx m 

J (a+ bx 2 ) n ~2b(n- l)(a + fcx 2 )"- 1 2 b(n - 1) J (a + bx 2 ) n ~ 1 ' 

15 f <fa _ 1 f dx b f dx 

Jx\a + bx 2 ) n a J x 2 (a + fcx 2 )"- 1 a)(a + bx 2 ) n ' 

EXPEESSIONS CONTAINING Va + bx. 

Also see Ex. 10, page 312. 



_ Va + bx b f dx 



f dx _ Va + bx b f 
J x 2 Va + bx ax 2a ^ 



17. fj Va + bx dx = 2Va~+Tx + atj 



xv a + bx 
dx 



xv a + bx 



INTEGRALS. 453 



EXPRESSIONS CONTAINING Vx 2 ± a 2 . 

Also see Ex. 7, page 312. 



18. ( — — = log ( x + ^ ± -^\ . See XXIV., XXV., page 181. 

J Vx 2 ± a 2 V a J 

n 

19. f (x 2 ± a 2 )2~^x = ^ ± <* 2 ) 2 ± ^«L f ( X 2 ± 2)2" 1 (?x . 
J » + 1 n+U 

20. f (x 2 ± arfdx = - Vx 2 ± a 2 ± — log (x + Vx 2 ± a 2 ). 

»/ A A 

21. f (x 2 ± a 2 )t dx = - (2 x 2 ± 5 a 2 ) Vx 2 ± a 2 + — log (x + Vx 2 ± a 2 ). 
./ 8 8 

22. fx 2 (x 2 ± a 2 )* dx = | (2 x 2 ± a 2 ) Vx 2 ± a 2 - ^ log (x + Vx 2 ± a 2 ) . 
»/ 8 8 



'I 



dx , x 



(x 2 ± a 2 )f a 2 Vx 2 ± a 2 



^ f x 2 dx = gVx 2 ±a 2 T-log(x + Vx 2 ±« 2 ). 

(x 2 ± a 2 )* 

25. f g2f?a; = X + l 0g (a; + V a ;2_ a 2^ 

J (x 2 ± a 2 )i Vx 2 - a 2 



27 



r — dte — = i log x . r — ^x__ = i sec _ 1 x 

-- _(. a 2)i a a + Vx 2 ^ - J 
C dx _ ^ Vx 2 ± a 2 



x(x 2 + a 2 )* a a+Vx 2 + a 2 J ^( x 2 _ a 2)S « « 



x 2 (x 2 ± a 2 )- 



28. a f ^ = _ ^g+g + -l-log «±vg ±g. 

' J x3(x 2 + a 2 )^ 2 «^ 2 2 « 3 

6 f ^ ^^ I 1 sec -i * 

"■>*(*-*)* 2a¥ 2as a 

29. q. f^ + ^^ = VgT p_ glog i±^±g. 

J x ° x 

r(x 2 -arfdx ,-„ = .a 

o. \ - <- = Vx 2 — a 2 — a cos -1 - • 

J x x 

so. f ^±<f^ = - ^±^ + iog( 8 + ^gr±^. 

J X 2 X 



454 DIFFERENTIAL CALCULUS. 



EXPRESSIONS CONTAINING Va 2 - x 2 . 
Also see Ex. 7, page 312. 



J w + 1 w + 1 J 



r x w c?x _ _ ^-Va 2 - x 2 (m — l)q 2 f x m ~ 2 dx 
J Va 2 - x 2 m m J Va 2 -^ 



33. r^v^^^= xw+lv ^^ 2 +-g 2 -f *■*» . 

J to + 2 m + 2 J y/ a 2 _ #2 

34. f dX dx=- ^* 2 -x 2 + m-2 f <fa 

J x m Va 2 - x 2 ( w - l)^™- 1 (to - l)a 2 J x m ~ 2 yj a i _ x a 

35 rVa 2 -^ 2 ^^ Va 2 -x 2 a 2 f dx 

' J x m (to — 2)x w ~ 1 m - 2 J x w Va 2 — x 2 

36. f (a 2 - x 2 ) *dx = - Va 2 - x 2 + 5- sin-i *. 
J ^ ' 2 -2a 

37. f (a 2 - x 2 )^x = f (5 a 2 - 2 x 2 ) Va 2 - x 2 + — sin-* -• 

38. f x 2 (a 2 - x 2 )* dx = - (2 x 2 - a 2 ) Va 2 - x 2 + ^ sin-* ?. 
J 8 8 a 

39. f — x2(lx = _gVa 2 -a; 2 + — sin-'-- 

it /V 4 CttA/ JO >f 1 4 "^ CM5 



r <%« _ « 41 r 

(a 2 - x 2 )^ « 2 ^« 2 - x2 ^ (a 2 - x 2 )^ vV - x 2 



a 



42. f *? _^_ Va 2 -x 2 j 43 C dx = li og . 

J a / o on o~ 0/ X •/ __ /• __o ..on ?T C? 



( a 2_3.2)£ « a+ V" 2 -X 2 



44 f <%g = Va 2 - x 2 1 lQ x 

^ ..., o ..„s* 2a¥ 2«3 



45 



X 3( a 2_ x 2^ 2 « 2 ^ 2 2 « 3 a + Va 2 - x 2 

*^ /=5— : r 9 „i_« + Va 2 -x 2 



. f (a 2 -x 2 ) dx = Va 2 - x 2 - a log 
J x 



J 



("'-^' a, = - -ME± = _ ai „-j ?. 



46. t ^' - ' ■ < fo != -JL2 ^ = -sin 



INTEGRALS. 455 



EXPKESSIONS CONTAINING V2 ax - x 2 , V2 ax + x 2 . 



[Here X denotes V2 ax — x 2 , and Z denotes V2 ax + x 2 .] 

47. a. J|? = siii-i^=^. b - |f = log(x + a + Z). 

48. a. fxdx=^^X+«- 2 sin-i^=^. 

J 2 2 a 

6. rZax=^±-^Z-^log(x + a + Z). 
•/ 2 2 

49. a. (Wcfe = -*'"-' Jr V 2 '» + 1 > a fx»-iX& 

J m + 2 » + 2 J 

J m + 2 m + 2 J 

fax X . m - 1 r dx 
. a. I = 1 I • 

J x m X (2 m -l)ax" 1 (2 m — I) a J x m ~ 1 X 

Jdx _ - Z m - 1 

x m Z~ (2 m - l)ax OT (2»i-l> 

"ax_ x m ~ l X . (2 m- l)a T 

X m m J X 

Z 



50 



h C dx - - z m — 1 C dx 

' J x m Z~ (2m - l)ax OT (2 m - l)a J x m ~ l . 

51 a C x m d% = x m ~ 1 X . (2m-l)a C x m ~ 1 dx 
J X m m J 



, r x m dx _ x m - l Z (2m-l)a C x m ~^ dx 
J Z m m J 



b . CL dx = *L_ »-« f_g_ te 

J X 7 " 



J x™ (2m- 3)ax™ (2m — 3)a J x m ~ l 

2? m-3 

(2m-3)ax m (2 m - 3)aJ x™- 1 

53. a. (xXdx = - Sa " ± gx ~ 2 x2 X + ^ sin-i ^?. 

J 6 2a 

5. fsZefo = - 3g8 - q *- 2 ^ Z + ^log(s + a + Z). 

^ 6 2 

54. . f*L = _i 6.f^ = _Z. 

J xX ax J xZ ax 

55. a . r^ = _x+asin-i^^. &. f £^ = z - alog(x + a + Z> 

»/ X a J Z 

56. a. f^ = - * + 3g X + 3q*8in-ig^g. 

J X 2 2 a 

& |x^ = x-_3a z + | a21og(x + a + z)> 



456 DIFFERENTIAL CALCULUS. 

57. a , r±^ = x+asin-i^-?. b. nL^ = Z+ alog(x+ a + Z). 

j x a J x 

58. a. r?<foj = -^-sin-i^^. b. (-dx = - 2 -^ +log(x + a + Z\ 

J x 2 x a J x 2 x 



59. a. (^dx = -^-. b. f4dse=- 
J x 3 3 ax 3 J x 3 



C dx _ x — a . f a"x _ x + a 
' J X 3 ~ a 2 X ' ' J Z 3 ~ a 2 Z 

f x a"x _ a; & C xdx 
' J X 3 ~ aj" ' J Z 3 



X 3 aX J Z 3 aZ 

EXPRESSIONS CONTAINING a + bx ± ex 2 . 

a . C ^ = 2 ■ tan-i 2 cx + b . for 6 2 < 4 ac 

J a + 6x + cx 2 V4 ac - b 2 V4 ac - b 2 



log 2ca; + 5-V 6 2 -4 ac < f Qr &2 > 4 ^ 



V& 2 - 4 ac 2 cx + & + V& 2 - 4 






5 (* <fo _ 1 j V& 2 + 4 ac + 2 cx - 6 

' J a + bx- cx 2 V6 2 + 4 ac V& 2 + 4 ac - 2 ca: + b 

63. a. r dx = J_ i g (2 cx + & + 2 Vc Va + 6a: + cx 2 ). 

^ Va + bx + cx 2 Vc 

b r ax = J_ sin -! 2c X - h 

J Va + bx — cx 2 Vc Vb 2 + 4 ac 

64. a. f Va + 5x + cx^x = 2 cx + b Va + 5a: + cx 2 

J 4:C 

_ 5 2 -4ac log ( 2 c» + & + 2 Vc Va + &x + ca: 2 ). 

8c2 

6. rVa + 6x-ca: 2 da: = :g^^Va + 5x-cx 2 + 62+4ac sin-i 2cx ~ b ■ 
J 4c 8 c* V& 2 +4ac 

eK . f a: ax _ VaTlJx - ^"^ 



l 



Va + &x + cx' 2 c 

^ log (2 cx + b + 2 Vc Va + bx + cx 2 ). 

2c^ 
ax _ _ Va + &x — cx 2 _6_ sin _! 2 cx — b 



J Va + t 



bx — cx 2 c 2 c* V& 2 + 4 ac 

N.B. Other algebraic integrals that are occasionally useful are given 
in Exs. 7-10, page 312, and in Exs. 4, 6, page 343. 



INTEGRALS. 457 

EXPONENTIAL AND TRIGONOMETRIC EXPRESSIONS. 

The most elementary of these are given in the integrals on pages 293, 301. 

66. a. fsmxcos"acfc = - cosW+lx - b. (*sin w as cosx = smW+la; . 

J ?i+l J n + 1 

67. «. | sin' 2 xc?x =^— ^sin 2x. b. | cos 2 xdx =-+ £sin 2x. 

CQ C • -, sin"- 1 ; x cos x n — 1 f . „ 7 

68. I sm"xdx = 1 \ sm n - 2 xdx. 

J n n J 

ca f -, cos" -1 as sin as re — If „ , 

69. \ cos n xdx = 1 \ cos n ~ 2 xdx. 

J re re J 

70 f ^ — 1 cos a; , re — 2 f <2x 

J sin n x re — 1 sin 71 - 1 ^ re — 1 J sin n ~ 2 x 

71 f oto 1 sin x re — 2 f (?x 

J cos"x re — 1 cos n_1 x re — 1 J cos n_2 x 

72. (*sec"x<2x = taQ asec»- 9 x + *^ f sec«- 2 xdx. (Cf. 71.) 
J re - 1 re - 1 J ^ y 

73. f coB60»gito = - cot x cosecn ~ 2x + VlIz1 f ooBee-'sdE. (Cf. 70.) 
J re — 1 re — 1J 

74. (*tan» a; dx = tan "~ 1 g _ ftan*- 2 xdx. 
J re — 1 J 

75. f cot" a; da; - - cotW ~ 1 x - (cot n ~ 2 xdx. 
J n — 1 J 

76. f S in^ a? COSn^^ = - silim - la;COSn+iag 

J m + n 

f s i n m- 2 a? cos w x dx. 



+ 



m + n 



77. f sin™ a? cos* xdx = sinm+1 ^ co ? n+1 * 
J m + 1 



m + n + 2 r s i n m+2a?cos w a?cfa\ 
m + 1 J • 



78. ( sin m a? cos™ a? eto 

J m + n 



sin™+i x cos** 1 x 

n-\ 



m + n 



j sin m x cos w ~ 2 x dx. 



79. jsin^cos^to=-^ inM 



n + 1 

+ m + n + f sin"* x cos n + 2 as d#, 
»+ 1 J 



458 DIFFERENTIAL CALCULUS. 

80. f sin tox sin nxdx = - sin ^ m ± n ^ x + sin ( m ~ n > - 

J 2 (to + n) 2 (to — n) 

01 f „^„ . „™ ~ ,j~ sin (to + n)x . sin (to — ri)x 

81. \ cos mx cos wx ax = * — ! — - — - l — 

J 2 (to + ») 2 (to - ») 

82. f sin tox cos nx <fa = - cos ^ m ± n > - cos < w = n > - 
J 2 (to + n) 2 (to - n) 



83. (* ^ = 2 tan-i (aP~^ tan ^ , when a > 5 

J a + b cos x Va 2 — b' 2 \ ' a + 6 2/ 



V& + a + Vb-a tan - 
log , when a < b. 



V& 2 - a 2 V6 + a - V& - a tan ? 

a tan - + 6 

84. f ^ = 2 - tan- 1 — 2 when a > 6 

J a + b sin x Va 2 - & 2 Va 2 - 6 2 

atan-+£-V& 2 -a 2 
log , when a<6. 



V6 2 -a 2 «tan^+6+V6 2 -a 2 



85. f ^ = J-tan-if 6taiia; V 

J a 2 cos 2 x + b 2 sin 2 x ab \ a ) 

86. f e- sin nsVfe = ^ sin nx ' n cos ^. (See Ex. 19, Art. 176.) 
J a 2 + w 2 

87. ( e«* cos rcx dx = e ^ n sin ^ + a cos "*). (See Ex. 6, Art. 176.) 




y - cos x 



y -= sin-\c y =eos~ x x 



459 





1 






/ 

/ 

/ 
/ 

/ 




71 

2 


0/ 


7T 
2 


2/ = ta 


37T 
2 

n a? 


/27T 57L 


X 






07? ' 

2 ^ — " 

27T 






37T 

2 ^ — " 

7T 






7T ^_______— ■ 




^^^0 


X 






2 ^ 




?/=tan" 


_37T 
2 



460 





■yn 
2 






y = sec l x 



461 



The Parabola £ 2 + y 2 =a ! 





The Cubical Parabola a 2 y ==x 3 



The Astroid or Four-Cusped 

1 2. 2 

Hypocycloid, x 3 + y 3 = a 3 




The Cissoid of Diodes 

o * 3 

2/ 2 =^TT 




Asymptote 



The Witch of Agnesi 



462 





The Folium of Descartes 
x 3 +y 3 =3ax y 



O X 

The Catenary 
y=f(e f +e- f ) 



Asymptote O X 

The Exponential Curve 

y=e* 




The Cycloid 
x=a (0-sin0), y=a (1-cos d) 






The Logarithmic Curve 
y-log- x 



Parabola 

o p 

n can - \l. 



The Cardioid 
r=a(l - cos 6) 



J 



4tf3 




The Lemniscate, rLa 2 cos 2 0, The Curve, r=a sin 20 The Parabolic Spiral 

r 2 = a' 



Asymptote 





The Spiral of Archimedes, r^a ( 



The Hyperbolic or Reciprocal 
Spiral, r Q <= a 




The Lituus or Trumpet, The Logarithmic or Equiangular 



r*0=a* 



Spiral, r= e ° " or log r= a 



464 



ANSWERS TO THE EXAMPLES. 



:>XKc 



CHAPTER I. 

Art. 4. 1. 45°, 0°, 63° 26' 4", 71° 33' 54", 75° 57' 49", 78° 41' 24", 
80° 32' 16", 82° 52' 30", 104° 2' 11", 99° 27' 44", 135°, 126° 52'.2, 110°33'.3. 
2. (.18, .033), (.29, .083), (.5, .25), (.87, .75), (5.72, 32.66), (- 1.07, 1.15), 

(- .35, .12), (- .18, .033), (- .09, .008). 3. [The latter part.] (a) - -; 

(b) 2s + l; (c) 3*2; (d) *; (e) ±£*; (/) M ; ( g ) lR ; (/,) _^E; 

?/ lb y lb y y a 2 y 

n\ b % 4. a. oo, ± .5774, ± .2582, 0, ± .4045, ± 1.8074 ; 90°, 30° and 

a 2 y 
150°, 14°28'.7 and 165°31'.3, 0°, 22°1'.4 and 157°58.'6, 61° 2'. 7 and 118°57'.3. 

b. 27, 12, 3, 0, 6.75, 18.75; 87° 52'. 7, 85° 14'.2, 71°33\9, 0°, 81° 34'.4. 
86° 56'. 8. c. oo, ± 1.4142, ± 1, ± .8165, ± .5774, ± .5 ; 90°, 54°44'.l and 
125° 15'.9, 45° and 135°, 39° 14' and 140° 46', 30° and 150°, 26° 34' and 153° 26'. 

d. 0, ± .1937, ± .4330, oo, ± .0945, ± .3034 ; 0°, 10° 57'.7 and 169° 2'.3, 
23° 24'. 8 and 156°35'.2, 90°, 5° 24' and 174° 36', 16° 52'.7 and 163°7>'.3. 

e. oo, ±.8661, ±.8183, ±1.25, ±.9139; 90°, 40° 53'.8 and 139°6'.2, 
39° 17. '6 and 140° 42. '4, 51°20'.4 and 128°39'.6, 42°25'.4 and 137°34'.6. 
5. Where x = ± 2.57 ; where x = ± 2.78. 

CHAPTER IT. 

Art. 12. 1. 35.2426 or 26.7574, 23.0186 or 21.1214, 3VsTnx + -^- 

+ 7sin 2 x + 2. 2. 68, 28, 14, 3 sin 2 x - 5 sinx + 21. 3. A4 ~ & x . 4. 18 + 

2-49x 

8Vx + x, 4 + Vx 2 + 2. 5. ay 2 + bxy + ex 2 , (a + 6+ c)x 2 , (a + & + c)y 2 . 

CHAPTER III. 

Art. 20. 1. (a) 22.977 ; (6) - 4.448. 2. (a) 21.22 ; (b) 40.42 ; 

(c) 161.58. 3. (a) .0047 ; (&) - .014. 4. (a) - .0035 ; (&) .0104. 

Art. 21. 3. 76.59, 22.24. 4. 212.2, 404.2, 538.6. 5. .80756, 

- .8023, - .60137, .5959. 

Art. 22. 4. (a) 2 x, 2 x, 2x; (&) 3x 2 , 3 x 2 , 3x 2 . 5. 4x 3 , 2 x + 4, 

-— , _l-3 + 4x. 6. 6t, m 2 -8--. 7. by b , ?«_8+-^-. 
x 2 x 2 £ 2 * 2 * ?/ 2 

Art. 26. 2. 2 7rr times, r being the measure of the radius ; 1.51 sq. in. 
per second ; 2.83 sq. in. per second. 3. .866 a times, a being the measure 
of the side ; 25.98 and 51.96 sq. in. per second. 4. 4 xr 2 times, r being the 
measure of the radius ; 9.425 and 37.7 cu. in. per second. 5. 5|^ mi. per hour. 

465 



466 DIFFERENTIAL CALCULUS. 

Art. 27. 3. 3x 2 dx, dx, 2 dx, 3 dx, adx, 2xdx, 14xdx, etc. 4. 1.6; 

1.681. 5. 42.2 ; 43.696. Ex. 5.03 and 9.425 sq. in. Ex. 1.3 and 2.6 sq. in. 



CHAPTER IV. 

Art. 31. 6x 2 + 14x-10, 2x-17, -2 a + 21. 
Art. 32. 4. (5 x 4 - 8 x 3 + 21 x 2 + 2 x - 2) dx, .... 

Art 33 1 8 ^ ~ 14 - + 6 x2 ) 16 ^- 21 ^ 2 -^ - 2 x 2 + 44 x - 96 

(3 x 2 - 7 x + 2) 2 ' (x 3 + 8) 2 ' (2 x 2 - 9 x + 3) 2 ' 

(3 x i - 14 x 3 + 6 x 2 ) dx _ 2 17 -8 

(3x 2 -7x + 2) 2 ' "" ' G °' 640' 245' 

Art. 35. 2. 4 *( 3 ' 2 " 4 ). 3. -i 4 ^- 

4 M- 17 3 x + 7 

Art. 37. 1.2 m-, 12 w 3 ^, 63 w^, 8 x 7 , 12 x 3 , 84 x 11 , 27 x 2 - 34 x + 10. 

dx dx dx 

3. 240 x(5 x 2 - 10) 23 , 120 x 3 (3 x 4 + 2) 9 , (432 x 5 + 300 x 3 - 168 x 2 + 448 x - 50) 
(4 x 2 + 5) 7 (3 x 4 - 2 x + 7) 4 . 4.-2 w 3 u', - 7 ir B «', - 11 w" 12 w', - 7 x~ 8 , 
-15x" 6 , -170X" 11 , -8x(x 2 -3)" 5 , -60x 3 (3x 4 + 7)" 6 , 15x 4 -21x 2 + 

4 * + 7 (2x 2 + 7x-3)~^ • ^ , _|(3x-7)"5, 6x-|x _2i -x^- 

3 V2x + 7 

2 ^ + 33 »\£ 6- v^ ^2-1 w ', V3 a/s-i, 5 V7 x^-i, 2 V5 (2 x + 5)^-1, 

V3(6 x + 7) (3 x 2 + 7 x — 4) n/3 ~ 1 . 7. — + c, and give c any three particular 

4 



constant values. 8. (In each of these expressions k is to be given any three 

x 6 1 2 - 

particular constant values.) — |- k, h k, -x 2 

6 x 3 

-2Vx + &. 12. 6x 2 + 34 x- 61, max™- 1 - «&x-« 



r 6 1 2 ^ 2 5 62 

particular constant values. ) — + k, - - + k, - x 2 + k, -x^ + k, - x 5 + - 

6 x 3 5 5 x 

i 4x -2a 

' (1-x 2 ) 2 ' (a + x) 2 ' 



12 ^-f^ 1 



VT+tf * 5 3 x 2 VTT^ («-&x 2 )^ (1-x 2 )* 

l mnx n - 1 (l + x n ) m ~\ 12 6x 2 (a + &x 3 ) 3 , x n ~ l 0- - x) n ~ l 

(1 - x) Vl - x 2 

[m -(» + »)*], <*~ 3 * . 14. a. f^t 4 ' ( f + 9 ay »V 

2V^^ if -ax «(3*/ 2 -2x 2 ) 

9x 2 y-8x- 14x*/ 2 -2y 3 -(x + a)y 2 , _* & 2 a 

14x 2 */ + 6x?/ 2 -3x 3 -16 2/ (a + */)(6 2 - a?/ - 2^/ 2 ) + ?/(x + a) 2 ' ?/' a 2 */ 
6. - |, f, f , - f • 17. ?/ = x 2 + k, in which k is an arbitrary constant ; 
y = x 2 + 1. 18. 5 ft. per second. 19. 10 mi. an hour ; 8f ft. per second. 
20. (4,8). 21. 3hr. ; 60 mi. 22. £ ft. per second. 23. 36° 52'.2. 
24. 36°52'.2. 



ANSWERS. 467 

(6x + 4)log a e 6x + 4 .434 (6 x + 4) 11 

' • 3 x 2 + 4a; _7' 3s 2 + 4x-7' 3a;2 + 4a ._7' ioiog e a' 

ii .29858. 2. i, .144765. 3. -=^- ? -X- I -, 1 . 

I-* 2 1-*' (1-sb)V5 Vx' + a 2 

— L_ , l + log x. 4. log (x 2 + 3 x + 5) + c, log c (x 3 - 7 x - 1), log Vfcx, 
xlogx 

in which c and k are arbitrary constants. (i7x. Write each of these anti- 
derivatives with the arbitrary constant involved in other ways.) 
6 ( \ - (2167 + 1877 x + 228 x 2 ) Vx + 2 (J)) 6(x 2 - 2) 

" 30(4x-7)t(3x + 5)t ' (.+ l)»(« + 2)«' 

91 x 2 + 475 x + 450 



00 



15(2x + 5)2(7x-5)3(x + 3) 



Art. 40. 1. 2xe* 2 , 2.303(10*), 2.303(6 x . 10 3 * 2 ), -1_ e^. 2. 2 e 2 ', 

2Vx 
2.303(2 «• 10' 2 ), 2^ 2 + 3 , 4.606 (10 a + 7 ). 3. e x x m ~ l {x + m), na* n • x"- 1 log a, 

- — , (1 - x) e-*, r _ 4 _ o , e* 2 f 2 - iy 4. i e 3 * + c, J e* 2 + c, 



(e x - l) 2 (e*+e- x ) 2 V x' 2 

| e^+i + c, c being an arbitrary constant. 

Art. 41. 2. (3x+7)* 2 f~2xlog(3x+7) + 3a;2 1, (3x+7) 2 *riog(3x+7) 2 

+ -•*-], as last, %(I=l5£*\ x* M .x»-i(nlogx+l), <* . e*, - ^-floga, 
3x+<J \ & I e\xj 

- - log a. 
x 

Art. 42. 1. — sin 2 w = cos 2 « . — (2 u) = 2 cos 2 w . — , 3 cos 3 w • Du, 
dx dx dx 

\ cos i w • u', f cos | m— , V cos V M ' -° M - 2 - 1> sin 2 x = cos 2 x • D (2 x) 
= 2 cos 2 x, 3 cos 3 x, \ cos \ x, 6 x cos 3 x 2 , 3 sin 6 x, 20 x 4 cos 4 x 5 , 

20 sin* 4 x cos 4 x. 3. 5 cos 5 t, t cos |* 2 . 4. 2 cos2 xsin 3x-3 sin 2xcos3x 

2 sin 2 3 x 

sin 2 x + 2 x cos 2 x, 2 x sin ( x + -^ + z 2 cos ( x + -V 5. 45° and 135°. 

6. Where x = *mt ± . 9553, in which n is any integer. 7. 63° 26' and 116° 

34'. 8. Where x = n-w — -, in which n is any integer ; 54° 44'. 1 and 125° 
4 

15'. 9 ; where x = mr + -, n being any integer. 9. n cos wx, wx n_1 cos x n , 

w sin*- 1 x cos x, 2xcos(l+x 2 ), wcos(wx + «), w&x"- 1 cos (a + &x M ), 

io :«« J i»»«-i» xcosx — sin x cos (log x) . „ / „\ i , sin e x 

12snv4xcos4x, , i — s — i, cot x, e z cos (e*) • log x H • 

x 2 x x 

10. (a) sin x + c, i sin 3 x + c, $ sin (2 x + 5) + c, \ sin (x 2 — 1) + c, in 

which c is an arbitrary constant. (b) \ sin 2 x + c, ^ sin (3 x — 7) + c, 

i sin x 3 + c, in which c is any constant. 



468 DIFFERENTIAL CALCULUS. 

Art. 43. 3. Where x = mr, n being an integer ; where x = (4»-l)- 
± • 485, 2 mr - . 485. 5 - - cot 0. 6. cot ' ; 60°. 7.-2 sin (2 x + 5), 

— 15 cos 2 5 x sin 5 x, 2 x cos £ — x 2 sin x, , — Cm cos nx sin mx 

(1 + cosx) 2 v 

+ w cos mx sin nx) , e cosx (l — xsinx), e ax (acosmx— m sin mx). 8. — cosx+c, 

— 2 cos \ x + c, — | cos (3 x — 2) + c, — \ cos (x 2 + 4) + c ; c being an arbi- 
trary constant. 

Art. 44. 3. 2 sec 2 2 w • Z)w, 3 sec 2 3 u • Du, m sec 2 mu • u', 2 wm sec 2 ww 2 • u' , 
2 sec 2 2 x, I sec 2 | x, ra sec 2 ?wx, 6 x sec 2 3 x 2 , 12 x 2 sec 2 4 x 3 , wmx n_1 sec 2 mx", 
6 tan 3 x sec 2 3 x, 12 tan 2 4 x sec 2 4 x, nm tan 71-1 mx sec 2 wix, f tan (f x + 3)sec 2 

(§x + 3), orcosecx. 4. tanx + c, |tan2x + c, |tan(3x+«) + c 

sinx 
6. When x is an odd multiple of - and dx is finite. 

Art. 48. 1. -2csc 2 (2x + 3), isec(ix+3) tan Qx+3), -3csc(3x-7) 
cot (3 x- 7), 5sin(5x + 2), wsec w xtanx. 2. -6 cot (3t + 1) esc 2 (3« + 1), 
sec 3 (i£- l)tan(i£- 1), - fese 2 § (t + 5) cot f(« + 5), -18«csc 2 9f 2 , 
14(7£-2)sec(7*-2) 2 tan(7 - 2) 2 . 

Art. 49. 2. nxn ~ 1 1 2 2 

Vl - x 2 »' Vl -2x-x 2 ' l+« 2 ' (l-x 2 )Vl-5x*' 
1 x sin -1 x 



| Vl + esc x. 4. sin -1 x + a, sin -1 x 2 + «» 



Vl — x' 2 Vl — x" 

\ sin -1 x 3 + a, in which « is an arbitrary constant. 

Art. 50. 3. - 2 "**- 1 , _2_, « . 

Art 51 1 2 2 dy 2x 3y 2 dy 2 4 

'l+4x 2 ' l+42/ 2 ax' 1 + x 4 ' 1 + ^dx 'l + 16£ 2 ' 

4 * 3 6x dx . 2 1 - x 2 1 1 



1 + £ 8 1 + 9 x 4 eft 1+x 2 1 + 3x 2 + x 4 VUx" 2 2(1 + x 2 ) 

a 3a 



2(a + 2x)Vx(a + x) a* + & 



7. tan -1 x + c, tan -1 x 2 + c, ^ tan -1 x 4 + c. 



Art. 52. 2. -^i. Art. 53. 2. 



x 4 -a* ^Vx*-1 VT^x 2 Va^x 2 * 2 +l 

Art. 55. 1. — - 

1 + x 2 

Art. 56. 2. (3 x 2 y 2 + 3) dy+ (2 x*/ 3 +2)dx, 3(y 2 -ax) dy+3(x 2 -ay) dx, etc. 

3 -J y - _-\fe -(^W^Y 71-1 ?/ tan x + log sin y 4 dx dy ^ 

V *x' \a) \y) ' log cos x - x cot y ' ' 2Vx 2Vy' 

|(^ + 4^V m (^^ + r^V (2/ tanx + logsin^x-(logcosx 

— x cot y) dy. 



ANSWERS. 469 

Page 77. 1. (i) 24 x 3 + 15 x 2 + 124 x + 55, (ii)a + 5 + 2x, (iii) (a + x)™"* 
(6 + x)»-> [m(& + x) + «(a + x)], (iv) (»>* - ws + W 6 - na) (a + q)»-i 



(x+6) 



a+l 



(v) (^^-y 1 , (vi) * , (vii) 



(l + *) w+1 (a 2 -*?)! (l + x 2 )t 



(viii) - v ;!^r -^ - , (ix)-^i + — -±_- , ( X ) 



2VxVa + x(Va+ Vx) 2 * 3 V Vl - x 4 / xVl - x 2 

a * + ^_4^ 2 28x 3 + 6x-17 -2a% -a 

VcP^x* 7x 4 + 3x 2 -17x + 2 a 4 -«* xVa^x 2 

(iv)secx, (v) — 3. (i)20x 4 cos4x 5 , (ii) — 7sinl4x, (iii)6sec 2 3xtan3x 

Vl+x 2 
(iv) 8sec 2 (8x + 5), (v) x m-1 (l + m logx), (vi) ^gx^sinP-^cosx? 

.«-!-«» ^- . IN- ^MN „«-/„;« «N ~™~ /^ cos (log wx) 



X 



(vii) n(sinx) n_1 sm(n + l)x, (viii) cos (sin x) • cos x, (ix) 
(x) ncotnx. 4. (i) — — , (ii) , (iii) — — — 6. (i) 



x 4 — 1 tan 2 x — 1 1 — x 4 e x + e _a 



00-1, m-=±=, ov) .,,,..., ■„... , w - v «- 2 -» 2 



y/l ~ x 2 cos'x + ii 2 sin 2 a; a + 6 cos x 

(vi) e ox sin" 1-1 rx(a sin rx + ?wr cos rx), (vii) log a • a 2 

x 2 

, ... N e x (2 - x 2 ) „ ,._ (x\ nx (, , sc\ ,.. N c ^/; c 

(Tm) (i - 8) vr^ - Ml) KJ i 1 + los ")' (B> i'r; 

(iii) x e V 1+xl ° ga; , (iv) e*V(l+logx), (v) x(* x ) -x*{ - + logx+ (logx) 2 } 

/ •% , 2 -i-i,i , oi x o ,-n ax + % + a .... x 2(x 2 + ?/ 2 )-a 2 
(vi) x z2+1 (l + 2 logx). 8. (1) — - ^ , (n) 



hx + by+f w y 2(x 2 + */ 2 ) + a 
2x?/ 4 ,. N 1, , - , , N cos x (cos ?/ + sin y) 

( m ) - .^,3 ,* „ ' (iv) -{msec (xy)-?/}, (v) 



4 x 2 ?/ 3 + cos ?/ ' v y x l sin x (cos y — sin y) — 1 

(¥i) *=i t (Tii) y-'yiosy (Tiii) ;»;'/ 9 . _2°** 

v y e2/-fx' v y x 2 -x?/logx' v y x( 1 + ray) (1 + logx) 2 



Vl^x 2 

10. (i) 2 y — |, (ii) 8 £ — 11, (iii) sec x, (iv) - cot z, (v) 

11. (i) (12 x 3 + 18 x + 5) (6 x 2 + 3) , (ii) (e tan * + 2 tan J) sec 2 *, (iii) gr. 

(iv) ^^ 12. (i) 90°, (ii) 73°41'.2, (iii) 90°, (iv)2°21'.7, (v)70°31'.7 

14. Speed of § in inches per second is 116.82, 225, 7, 319.18, 390.9, 436 
451.39, 390.9, 225.7, respectively. p. 419 

CHAPTER V. 

Art. 59. 2. See answer Art. 4, Ex. 3. 4. (i) ± 1, ± i, 45°, 135° 
26° 34', 153° 26'. (ii) 2, 63° 26'. (iii) - f, 146° 19'. (iv) - 1, 135° 
(v) 1|, 56° 18'. 6. (vi) 1|, 56 Q 18'.6. (vii) 1£, 56° 18'.6. (viii) -.6667 
146° 19'. 

Art. 61. 2. y=x — 12, 2y + x + 6=:0, x + y = 0, y = 2x-lS. 



470 DIFFERENTIAL CALCULUS. 

3. y + 2. 0056 x + 2.19-.. = 0, y = 4.6056 x- 10.6 •••, 2.6056 y = x + 14.6 ..., 
x + 4.6056 ?/ = 53.45 • ••. 4. (i) Tangents : y = x + 2, x + y + 2 = 0, 2y = 
x + 8, 2y+x+8 = 0; normals : y +x = 6, y = x — 6, y + 2 x = 24, y-—2x 

- 24 ; (ii) y = 2 x - 8, 2 ?/ + x = 24 ; (iii) 3 y + 2 a = 13, 2 ?/ = 3 x ; 
(iv) x + 2/ = 6, » = ?/; (v)2?/=:3x-3, 3?/ + 2x = 15; (vi) 2 y = 3 x, 3y 
+ 2x = 13; (vii) 2*/ = 3x- 10, By + 2 x = 24 ; (viii) 3 y + 2 x = 24, 
2y =Sx~10. 

Art. 62. 1. The lengths of the subnormal, sub tangent, tangent, and 

normal, are respectively : (1) 3, 54, 6f, 5; (2) 4, 4, 5.66, 5.66; (3) ~^1' 

a 2 

- a2 ~ Xl<2 , — V(a 2 - Xi 2 ) (a 2 - e*xi*) , & Vfl 2 -e 2 Xi 2 ^ g bdng the eccentricity . 

Xi Xi a 

(4) sinxi, cosxi, tan Xi, tan xi Vl -f cos 2 Xi, sin x\ Vl + cos 2 x\ ; (5) ?/i 2 , 1, 
Vl + «/i 2 , yiVl + ?/i 2 . 2. Where x is 7 ± 2 Vo. 3. Infinitely great. 

6. xxi"^ + yy{~* = a*. 7. xx-T^ + yyf^ = <A 8. a sin 0, 2 a sin 2 - tan -» 

2 2 

2 a sin -, 2 a sin - tan -. 12. 90°, 0°, cot" 1 4* i.e. 32° 12'. 5. 

2' 2 2 

Art. 64. 1. (1) a, a0 2 , a Vl+^, r Vl + 2 ; (2) — , 2r0,^V40 + 0-i 

aV0(l + 4 2 ); (3)--, - a, - Va 2 + r 2 , -v^T^; (4) wafl"- 1 , ^^, 
a a n 

a^- 1 Vn 2 + 2 , -Vw 2 + 2 . 3. ar, -, rVl + a 2 , -VI + « 2 . 4. (a) \p = 
n a a 

34° 55'.2, = 74° 55'.2 ; ^ = 50° 41'.9, 120°41'.9 ; (5) ^ = 26° 33'.9, 

= 55°12'.8. 



Art. 65. 1. In feet per second: 0, 4; 2.828, 2.828; 3.57, 1.79; 3.77, 

1.33. Solution for x = 2 : Where x = 2, the tangent to the parabola has 

a slope 1. Accordingly, the moving point is there going in a direction 
which is at angle 45° to the x-axis. Hence, the speed of the x-coordinate 

(i.e. <te\ - ^ x cos 45° = 4 x — ; also ^ = 4 x — .1 2. 20 and 22.36 ft. 
V dt J dt V2 dt V2 J 

per sec. Suggestion : Differentiation with respect to the time gives 

2 yty = 4 — .1 3. .399 and - 9.97 ft. per sec. ; 9.7 and - 2.425 ft. per sec. 
J dt dt A 

4. 442.82 and 161.2 ft. per sec. ; 199.15 and 427.08 ft. per sec. 5. (1) (2, 8), 

(- 2,-8); (2) (1, ^), (- |, - sf 6 ); (3) 300. 



ANSWERS. 471 

Art. 66. 3. 25.1 cu. in. ; ^. 4. 4 irr 2 • Ar ; 50.3 sq. in., 502.7 cu. in. ; 

To-rn 2 2 oo^ dh>- 5 - L35 sc l- in - 5 7 : 5 approximately. 

Art. 66a. 2. (1) - 1, - 1, § ; (2) - |, - |, - 1 ; (3) 2, 2, 3, 4; 
(4) - i, - i, 3, - f j (5) 2, 2, - 3, - 3, 1. 3. n n r n - 2 = 4p»(n - 2) w ~ 2 . 

Art. 67. 4. 1.6, .4. 6. ir 2 ,i.e.2^ 2 ; .0048, .035. 7. .0349, 0, .0025. 

9 J a + X , \l a + x ; Ife. 10. 2.41, .1. 11. avT+1 3 , ly/WT7\ 
> x * a *x a 

12. .078. 14. ttx 1 , ttx 2 . 15. 5.03, 10.05. 18. 10.37, 5.06. 19. J a ' 2 ~ e2 f 

* a- — x 2 

& Va 2 - x 2 7r6 ' 2 ^ 2 ~ x ^ , ^ Va 2 - e 2 x 2 e being the eccentricity. 20. - 
a a 2 a f 

a, ?• cosec a, V2 ar. 

CHAPTER VI. 

Art. 68. 1. (i) - 2 , 7NO ; (ii) 8 + 1 + -i_ ; (iii) 



fl + x 2 ) 2 ' x 3 4v /^' (l-sinx) 2 ' 

,(l + log^ + ^. 2 .(i)_^ i( ii)^ 3.(,)^±^ ; 
^a ■+■ x j biu x (1— x 2 )^ 

24(l-10x 2 + 5x*), (i) _4J + 8e * sina , 6 . (i) _ _*. 

k ; (1 + x 2 ) 5 w x 2 v J w a 2 ?/ 3 

M - n?aV 8 - (i) - L4) " 2 ' 66; (ii) h ~ h ' 9 - l —e> ~ h 

y. l +*y) 4 a sin* - 

12. 24 x. 13. -^ = |x 2 + 2 x + Ci, v = |x 3 + #' 2 + CiX + c 2 , in which c x and 

dx 
c 2 are arbitrary constants. 14. 3 y = x 3 — 9 x + 19. 15. ?/ = 4 x 2 + x. 
16. (2) '— | ft. per sec.' per sec. 17. In ' in. per sec' per sec. : (i) 1152 7r 2 , 
(ii) 768 71- 2 , (iii) 384 tt 2 , (iv) 0. 18. s = \gt 2 + c\t + c 2 . 19. 15.5 sec, 
3881.9 ft. 20. 30V0 sec- 
Art. 69. 2. e x , a z (loga) n , a n e ax , b n a hx (log a) n . 4. cos(x + — 

ansinfax+^Va-cosfax+^V 5. (-l)"^-!)!, (-l)*^- (n-l)l. 

V 2 J V 2 y x" (sc — 2)» 

g (- \) n n ! (— l) w rc ! ; 2- n\ (- l) w ac"(m + n - 1)! 

x n+l (1 + x)^ 1 ' (S-*)^ 1 ' (& + cx)™+» 

7 . w! { (-D" + I I n! /_L_ ( Ll )" I. 

l(l + x)»+i (l-x)" +1 J 1(1-sb)»+i (l+x)^ 1 / 

Art 71 2 q + & cos A , _ & + a cos 6 
&sin0 & 2 sin 3 

Art. 72. 2. (x 4 -120x 2 +120)xsinx-20(x 2 -12)x 2 cosx. 3. (x+w)e*, 
2»~ 1 (/i + 2x)e 2 *. 

Art. 73. 3. (1) y' = xy"; (2) x 2 y" + 2y = 2xy> • (Z) y' + 2xy" = 
(4) (x 2 -2y 2 )y 2 ' -4xyy' - x 2 = 0; (5) yij' = x(yy" + y 2 '). 4. (1) ?/' = 
(2)y = xy'; (3)y" = 0; (4) y" = y ; (5) y" = m 2 y; (6)y" + m 2 y = 
(7) y" + m 2 ?/ = 0. 5. y 2 (l + ?/ 2 ') = r 2 ; x 2 (l + y 2 ') = r 2 2/ 2 ' ; {1 + ?/ 2 'p = ry" 



472 DIFFERENTIAL CALCULUS. 

CHAPTER VII 

Art. 76. 4. A minimum ; neither a maximum nor a minimum. 8. See 
Ex. 3. 12. See Ex. 3. 13. (1) Min. for x = I ; max. for x = - 2. (2) Min. 

at ~ 1 ~ a/7S ; max. at ~ 1 + ^ 78 . (3) Max. for x = ; min. for x= 1-V ^ ; 

O D x'Ji 

min. for x = -^- ; f or x = 2, neither a max. nor a min. (4) Max. for 

12 ' w 

x = — 1 ; min. for x = | ; neither a max. nor a min. for x = 2. (5) Min. for 

x = 4. (6) Max. when a; =— 4, and when x = 3 ; min. when x = — 3, and 

when x = 4. (7) Min. for x = 16 ; max. for x = 4 ; neither for x = 10. 

(8) Max. for x = — 10 ; min. for x = — 2; neither for x = 2. (9) Min. 

value is + -, i.e. + .3678. (10) Max. when x = e. (11) Max. value = 8; 
e 

min. value = 2. (12) Max. or min. when sin x = Vf according as the angle 

x is in the first or the second quadrant. (13) Max. when x = cotx. 

16. (ay/2, aVl). 

Art. 77. 7. Each factor = Vthe number. 8. -• 9. A square. 

10. (i) (a J + 6=0 2 . (ji) a + 2Vab + 6 ; (iii) 2ab. 11. Let the perpen- 
diculars drawn from A and B to ilfiV meet MN in i? and S respectively ; 
then (1) BC = CS; (2) BC = f ^ ' BS - 12. (i) f r; (ii) f r. 13. 19°28'. 

14. (i) Vol. = .5773 vol. of sphere; (ii) height=rV2. 15. (i) Vol. ^^-rrcPb ; 
(ii) height = i b. 16. 1. 17. 2 C , i.e. 114°35^29".6. 18. Vf a. 19.1:2. 

22. Ii times the rate of the current. 23. -^ d, — <Z. . 24. (a* + 53) 2> 

o o 

25. -«-. 

V2 

Art. 78. 1. (1) (0, 0) ; (2) (3, - 3) ; (3) (f, W) 5 (*) (2, |) ; 
(5) (± — z 1 -); (6) where x = 0, and where x = ± VS ; (7) where x = 0, 

and where x = ± 2 V3. 2. (1) Where x = — ; (2) where x = — ; (3) where 

5 4 

fe = ±-7=; (4) (c.6); (5) {c,m); (6) (&, ^fV 
V3 V a 2 J 

CHAPTER VIII. 
Art. 79. 2. 3 x 2 + e* sin #, 4 ?/ + e x cos ?/ — cos z sin ?/, 6 z — sin 2 cos y. 

3. (a) ^ and -^ (6) ^ and ^* ; (c) -=^ and -46^ 
4V119 5V119 3V89 6V89 3V47 4V37 

respectively. 

Art. 81. 3. Increasing — - : units per second. 4. Decreasing = 

units per second. 20V119 5V89 



ANSWERS. 473 

Art. 82. 3. .030 ; .036011. 4. (i) x dy - ]ld:c ; (ii) y x logy- dx+xy*- l dy; 

x 2 + y 2 

(iii) yx*'- 1 dx + x'logx- dy ; fiv) 2 oVe + log sc • <fy ; (v) u f^^da: + ^^ • 

x \ x y J 

5. .025. 6. 2.2; 2.37. 7. .0017. 8. xv*- l (yz dx + zxlogx • dy + xy\ogx • dz). 
Art. 83. 3. 4.72 sq. in. 

CHAPTER IX. 
Art.90. /3,W-^W^V. 



Art. 93. l.{/'(/ / / "-0'/'")-/ // (/V"-07")}- ; -/ 5/ - 2. -4asin^. 
3. — a. 4. — (a 2 sin 2 + 6 2 cos 2 0) 2 -f- a5. 



Pagel47. 1. (i)f?_2j,§*=0. (ii)f?+f?=0. iff-iff" 

= cos^. 3. (i)^+„ = 0. (ii)fl + J/ = 0. (iii) ^=0. (iv)^ 

fty 2 dt~ dt 2 dz- 

+ a' 2 y = 0. (v)g + y = 0. (vi) g-P 2^4- y = z. 4. (i) tan* ; 

(ii) — 3 sin 4 1 cos t ; 3 sin 5 £ (4 — 5 sin 2 1) . 



at cos 3 1 

CHAPTER X. 

Art. 95. 1. (1) First order at (1, 1); (2) second order at (2, 8). 
2. y = 5 x 2 — 6 * + 3. 3.-1. 4. ?/ = 3 a 2 - 3 x + 1. 5. y = x 2 — 3 x + 3. 

Art. 96. 1. 5.27 and (- 4, f); 2.635 and (- f, -V). 2. B = 145.5 ; 

(-143, 20 T V). 

Art. 100. 1. The curvature of y = x 3 is one-half that of y=6 x 2 — 9x+4. 

2. — ; 2? =-88.4; (-87.5, -12.5). 

125 

3 

Art. 101. 3. 2 (P + *y 2 ; (2p + 3a;, — ^-\; 2p and (2p, 0). 

p h \ 4P 2 J 

3 



4 Jg = _(^+^^ = c^-^^ Centreat («!^!,3, .-tn»vy 



m ^^; (l. + Jt, ti + ^V (3) *; (.-i^*,«i 

2 a- \ 2 a 2 2 a 2 / a V a 



(4) - 3(aa#) 3 ; (x + 3 Vx^, y + 3 Vx 2 y). (5) 3 a sin cos ; (a cos 3 « 

3. 1 

+3«coe*sin 2 *, asin 3 J+3asin«cos?«)- (6) ( 4 « + 9 a) 2 ^ . / E-f— , 

6 a \ "a 



474 DIFFERENTIAL CALCULUS. 

±V + t— ) ■ (7) - 2 a ; (a, - ] a). (8) ± 4 a sin* ; (a . d + sin 0, - y). 

K / 3 3 ^ 

6. (1) ( X + VY . (2) ( a * + 9a:4 ) 2 , (3) C sec-. (4) 4 a. (5) 2 a cosec* yh. 

(6) (fl 2 sin 2 + fr 2 cos 2 0)2 (7) _ (a? tang + & 2 sec2 0)* aS ec 2 f, 

2 «6 a& 

i.e. sL . 

a 

Art. 102. 1. (1) a; b. (2) 2 r v ^ . (3) f vT^ 7 . (4) -^-. (5) ±— . 

Va a 2 3 r 

(6^ rvT+tf. (7) ±"(1 + »')*. (8) ± ^n-i (w -2 + . 02) l 

V ' W 2+02 W =«= H( ^ + 1)+0 -2 

Art. 103. 3. (1) (ax)* -(6y)* = (a 2 + 6 2 ) 3 . (2) (z + ^-(x-t/)l 
= (4 a) 3 . [Suggestion: Show that « +/3 = ^ + -V, cc-° 



2\x a/ 2 \x a 

2 2 2 

and deduce therefrom.] (3) (x + #) 3 + (x — ?/) 3 = 2 a 3 . 



CHAPTEE XI. 

Art. 108. 2. i J. 3. The tangent at the middle point of a parabolic arc 
is parallel to the chord of the arc. 4. — 3 ± V^f ; find the abscissa of the 
point where the tangent is parallel to the chord joining the points whose 
abscissas are 3 and 4. 

Art. 109. 1. -3.69-.., .51 •••, 3.18 .... 2. -4.03293, 1.2556, 1.77733. 
3. 2.858,-3.907. 4. 2.34. 5. 2.046. 6. -3.806. 7. 2.129. 

8. 2.215, -.5392, -1.676. 9.3.693. 10. 1.4231,-0.6696. 11.2.195823. 



CHAPTER XIII. 

Art. 123. 5. (1) x 2 + y 2 = a 2 . (2) b 2 x 2 + a 2 y 2 = a 2 b 2 . (3) 4 ay 2 + bxy 
+ ex 2 = 4 ox - b 2 . (1) 4xy + a 2 = 0. (5) 4^/ 3 = 27 a 2 x. (6) (x - a) 2 
+ (y- b) 2 = r 2 . 6. (1) x 2 + if- = a 2 . (2) x 2 -y 2 = a 2 . (3) (ax)* + (&*/) 3 
= (a 2 - 6 2 ) f. 7. (1) The lines x ± y = ; (2) 27 cy 2 = 4 x*. 10. A parabola ; 
y 2 =4: ax if the fixed point be (a, 0) and the fixed line be the z/-axis. 

Art. 124. 3. 4 xy = a 2 . 4. x^ + y~ 3 =a 3 . 5. x^ + ?/ 3 = a*. 

Art. 126. 4. (1) x=a. y = b. (2) a; = 2. (3) y + 3 = 0, 2a*+ 3 = 0. 

(4) y+l=0. 5. (2) (2. |). (3) (-f, -3), (- f, -J£). (4) (-|, -1). 
8. (1) s = 0, y = 0. (2)» = 2a. (3)y = 0. (4)x = ±a,i/ = i&. (5) y=0, 
x = a. (6) x = 0. (7)^ = 0. (8)^ = 0. (9) x = (± 2 » + 1) -, in which 

n is anv integer. 



ANSWERS. 475 

Art. 127. 2. bx±ay = 0. 5.(l)y=x. (2) x +y = 1, x- y = 1. 

(3) *=2, y+3=0, 2(y-a)=:5. (4) a:=y±l, sc+y =±1. (5)6*/ = 3x + 2. 

Art. 128. 2. (1) Lines parallel to the initial line and at a distance 
± nair from it, n being an integer. (2) The line perpendicular to the initial 
line at a distance a to the left of the pole. (3) The two lines which are 
parallel to the initial line and are at a distance 2 a from it. 4. r sin (6—1) =1 ; 
r = l. 

Art. 131. 3. (1) Node at origin; slopes there are ±1. (2) Cusp at 
(— 3, 1) ; slope there is 0. (3) Cusp at (2, 1) ; tangent there is parallel to 
the y-axis. (4) Double point at (0, 0) ; slopes of tangent there are 1, — f. 
(5) Cusp at (1, 2) ; slope of tangent there is 1. (6) A conjugate point at 
(3, 0). 

CHAPTER XIV. 

Art. 136. 1. 50 ft., N. 53° 8' E. 2. 51.96 ft., W. 3. 58.8 ft., 

N. 16° 3' E. 4. 9.39 ..., 3.42 ..., and 5.77 ... miles respectively. 

Art. 137. 1. 9.83 ... and 6.88 ... ft. respectively. 2. 25.5 ft., 43° 17' 
(nearly) to the horizon. 3. 228.8 ft., 377.3 ft. 4. 258.3 inclined at an 
angle (— 63° 27') to the given displacement. 

Art. 138. 1. 2.236 ft. per sec, E. 26° 34' S. 2. 20.47 miles per hour, 

N. 42° 3' E. 3. At an angle 60 c to the river bank. 4. 11.37 and 3.84 

miles per hour, respectively. 5. 21.56 mi. per hour toward the south, 

24.18 mi. per hour toward the west. 

Art. 139. 2. 4 ft, per sec; 3.79 ft. per sec; 3.79 ft. per sec; 3.84 ft. per 
sec; 21.21 ft. per sec 3. Decreasing 9.7 ft. per sec; increasing 8.77 ft. per sec. 

,7a ,7a 

4. (a) a sin ft — (in which — is the rate at which the radius vector is re- 

dt dt 

df) ft rift 

volving) ; Cb) a (I — cos 6) — ; (c) 2 a sin - — in a direction making an 
dt 2 dt 

ft ^i ft 

angle - with the radius vector (Le. - — with the initial line). At the points 

z Z 

(1), (2), (3), (4), the values of («), (6), (c) are respectively, in inches per 
sec: (1) 5.236, 5.236, 7.405; (2) 4.53, 2.62, 5.236; (3) 4.53, 7.85, 9.07; 

(4) 0, 10.47, 10.47. 

Art. 140. 1. 14.8 mi. per hour, N. 37° 50' E.; 4.9 mi. per hour, per 
hourX. 37° 50' E.; 9.08, 11.7, 3, 3.87 mi. hr. units. 2. 98.9 ft. per sec. 
per sec; 58.1 ft. per sec per sec 

Art. 141. 2. — a cos 6 [ — , in which 6 denotes the angle from the 

horizontal diameter to the radius drawn to P. 3. (a) 7047.75 ir 2 ft. per 

sec. per sec; (&) 3169.5 tt 2 ft. per sec. per sec. 4. 10.89 ft. per sec. per sec 

5. (a) 6 in. per sec. per sec; (6) 2 in. per sec. per sec; (c) 6.32 in. per 
sec. per sec, in a direction inclined at 71° 34' to the normal. 6. (a) 2 in. 



476 DIFFERENTIAL CALCULUS. 

per sec. per sec; (6) 1.61 in. per sec. per sec. directed toward the centre of 
curvature for P; (c) 2.56 in. per sec. per sec. in a direction making an 
angle (— 38° 50') with the tangent at P. 7. Wholly normal, 1.61 in. per 
sec. per sec. 9. Vel. =4.4 ft. per sec; at = 8.383 ft. per sec per sec. ; 
a n = 4.84 ft. per sec. per sec ; a = 9.68 ft. per sec. per sec 

CHAPTER XV. 

Art. 146. 3. (1) Convergent. (2) Convergent. (3) Divergent. 

(4) Divergent except when p > 2. (5) Convergent if p > 2. 4. (1) x < 1, 
convergent; x>l, or£ = l, divergent. (2) Absolutely convergent if 
x' 2 < 1, divergent if x 2 = 1, divergent if x 2 > 1. (3) Absolutely convergent 
for all values of x. (4) x < 1, orse = l, convergent; #>1, divergent. 

(5) Same as in Ex. (4). (6) Same as in Ex. (3). 

CHAPTER XVI. 

7,2 7,3 

Art. 150. 5. (a) cosx—h since cosaH sin £+•••; (b) cosh 

2 ! 3 ! 

x 2 x z 
— x sin h cos h -\ sin h + ••-. 

2! 3! 

Art. 151. 4. e + e(x-l) +-^-(z-l) 2 -f .... 

Art. 152. 10. (1) l + |i + ^ + ^! + ... ; (2) | + g + | + .... 

12 . (1) c + x + ^_^_2i- 5 _2^ + 2B^ +> log* + (5-«) 

w 2 4! 5! 6 ! 8 ! w a 

, &s-a3, &»-<,», _ (3)x __^ + a* 



2 • 2 ! 3 • 3 ! 1-3 1 . 2 • 5 1 • 2 • 3 • 7 



CHAPTER XVII. 

Art.162. 2. (a) 6 X + y 4- 3 s = 19, ^-^ = ?/ - 4 =^=i; (6)3*- 

6 o 

6^ + 7z + 19=0, 2x+?/ + 2 = and 7x-3s + 27=0; (c) 4 * + 8 y 

-32 + 6=0, 2x-y = 20 and 3 as + 4 z = 32 j (d) 4 a; - 18 # - z = 31, 

9iK + 2 y = 12 arid x + 4 3 = 126 ; (e) 3 x + y + 2 z = 0, a; = 3 y and 2 y = z ; 

(/) 2x + 2/ — 4^=4, x = 2y and 4 y + s = 20. 

Art. 163. 2. 2x + 12 y -9z + 48 = 0, Qx— y = 32 and 9 a; +22 = 78; 

(a) 2 a + 12 y — 9 z + 48 = and 3 x - 2 y + z = 22, 6 x + 29 y + 40 z = 632 ; 

(6) 2 x + 12 y - 9 s + 48 = and 4a: + y-3z + 8=0, 27 x + 30 y 4- 46 z 

= 834. 3. x-2?/-2 + 5 = 0, x=* + 3and?/ = 22; (a) a; - 2 y 

- 2 + 5 = and 7 x - 2 y - z = 25, y = 2 z ; (&) x - 2 */ - 2 + 5 = and 

2x + 3y+2 = 24; ar- -3y+7« = 7: 4. 8x - 27 y - 21 z = 122. 



ANSWERS. 477 

3a; + s = 15 and 8 y — 9 s +-75 = ; (a) 8 a - 27 ?/ - 24 z - 122 and 3 x - 

2 y -3s = 15, 33 £ - 48 «/ + 05 s = 615 ; (6) 8 x - 27 y - 24 a = 122 and 
¥ + 2 y + 4 z - 4, 60 x + 56 y - 43 s + 225 = 0. 5. 13 x + 30 y = 198 and 
32 y + 39 z = 596, 90 x - 39 y - 32 z = Q. 6. y + 4 x = 24 and 9 x = z + 43, 
x—±y +9z = 7. 

CHAPTER XVIII. 

Ait. 167. 3. ?/ = ^ 5 ?/ = x 3 - 347 ; y = x 3 + 514 ; y - h = x 3 - /t 3 . 
4. y = 4'x + c ; ?/ = 4 x ; y = 4 x — 5 ; y = 4 x + 29. 5. y = 4 x 2 + c ; y = 4 x 2 ; 
y = 4 x f _ 2 j y = 4 x 2 - 13 ; y = 4 x - 62. 8. 16 £ 2 ; 64 ; 256 ; 400 ; 16 t 2 + 10, 
etc. ; 16 £ 2 + 20. 

Art. 168. 3. f. 4. 2; 0. 5. 4; 0. 

Art. 170. 4. («) 2 y = x 2 , Q y = x 3 , 24 y = x 4 ; (5) ?/ = x 2 + 5 x, 6 ?/=2 x 3 
+ 15 x 2 , 12 ?/= x 4 + 10 x 3 ; (c) ?/ = 1 — cos x, y =x — sinx, 2 y = x 2 +2 cosx — 2 ; 
((7) i/ = e z — 1, y = e x — x — 1, 2 y = 2 e* - x 2 - 2 x - 2. 5. y = 1, y =2, 

y = cos x, y = e x . 

CHAPTER XIX. 

Art. 174. 9. ix 8 + c, T 6 3X 73 + c, ^ 2 r x 41 +c, - § x~ 18 + c, -^x^+c, 

— + c, - — + c, 4 x^ + c, a/" +1 + c, | x^" + c, 4 x^" + c, 8 Vx + c, 

2 x 2 x 4 V2 + 4 

-^ + c, --A- + C. 10. i v 4+c, -V^ + c, ^4+c, 12s 4 + c. 

y^ 14 X 4 2 W 4 

m+n »i+3 , 6+n — - 

11. _«»_«— + c, A-«T + jfc) 4l,X +c ™ s 
??i + ?i m + 3 6 + n £ + 

log c(s + 2) 2 , -ilogc(7-x 6 ), logc(4£ 2 -3£ + 11). 13. e* + c, fe^-fc, 

4 1 10 2x 

2 e* 2 + c, he, \-c. 14. — icos3x + c, 4 sin 7 x + c, | tan 5 x + c, 

log 4 2 log 10 

— cos (x + a) + c, i sin (2 x + a) + c, f tan [ — +- ) + c 15. \ sec 2 x +c, 



fseefx + c, sin-^ + c, isin-^+c, |sin- 1 5x+c, fsin-^ + c, log(v+ Vl + v 2 ) 
+ c, |tan _1 r 2 + c, tan -1 2 x + c, sec^ + c, sec -1 3 x + c, J sec _1 x 2 -f c, 
i vers -1 3 x + c or -i sin~ 1 (3a! — 1) + c, ivers _1 4x + c or ^ sin -1 (4x— 1) + c. 

16. f-f^+16* + c, a 4 x + Jf «%t + fi a l x l + _4_ .¥ + C5 » e i x +Cj 
o m 

- sin ax cos nx + c. 

[In the following integrals the arbitrary constant of integration is omitted.] 
Art. 175. 11. £sin<5x, t ^^ (3 + 2 tan 2 x) , -^tan(4-7x), - J e~ 2x . 

12. log(x + l)+ 4a; + 3 g+3x+31ogx-l, |(x+2)*(x-8), T \(x-2)3 

& \X-\- 1 ) Z X 



478 



DIFFERENTIAL CALCULUS, 



8. 

(2* + 3). 13. t(*+o)*i 5(m 8 + w nX)5 > -fV3^7~^, 1(4 + 52/)'. 

14. l e m+ nx ^ _ 45 ~ 3x log (tan -1 aj), -cos (log*). 15. A(*-l)*(3f + 2), 
w 3 log 4 

S 5. i 

— (a + by) 3 , f (m + s) 4 , f sin | *. 16. a sin* (3 -sin 2 *), a tan* (tan 2 * + 3), 

| cos 3 * — cos * — i cos 5 *, n tan [ - ) • 17. — } log (3 + 7 cos *), 

-i log (9 -2 sin*), - | V4 - 3 tan *, J-sin-i f ^3tan* \ i 8 . V^fl 2 , 

5 3 i VS \ V7 / 

-|(a 2 -* 2 )a, i( a 2 + * 2 ) 2 , — =£=• 

Va 2 - * 2 

Art. 176. 7. ^(ax-1). 8. - (* + 1) e~*. 9. ae"(* 2 - 2 a* + 2 a 2 ). 
a 2 

10. * lo g * - * . 11. i* 2 (log*-|). 12. i* 3 (31og*-l). 13. * tan -1 * 

— log Vl -f * 2 - 14. |(1 + * 2 ) tan -1 * — \ *. 15. 2 cos * + 2 * sin * — * 2 cos *. 
16. e x [x m - mx m ~ l + m(m - l)* w " 2 -... + (- l)«-i w(m - 1) ••• 3 • 2 • * 
+ (— l)»vf»!]. 17. — A* cos 2* + | sin 2*. 18. - Vl — * 2 • sin" 1 * + *. 

Art. 177. 7. — ttin-i ?-tl . s i n -i «j^§ . i og ( x + 3 + V* 2 + 6*+10). 

2V2 2V2 V26 

8. il og Z-±£; s i n -i 2a; + 5 ; log (2 * - 5 + 2 V* 2 - 5 * + 7). 9. — 

1 - » V53 V33 

log 2* + 5-Vg. JL log ^ + 5-V6T , i log(8a; ,3 + 4V 4* 2 -3*+5) . 



2 * + 5 + V33 V61 2 * + 5 + V61 



10. _l_tan-i^^ 

\/7l a/71 



I sin 



_t 8 * + 5 . 
13 ' 



1 1 V137 + 5 + 8* 



V137 Vl37 - 5 - 8 * 



11. vers- 1 - and sin" 1 ^ 4 -; I vers" 1 — and ism" 1 — — -: 1 sec" 1 — • 

4 4 ' 2 9 2 9 ' 25 5 

12. isec- 1 ^-^; - 1 (*V9 - * 2 + 9 sin" 1 ^ ; - 2 ^ tt. 13. *V9 - * 2 + 9 sin" 1 ?; 

2 \ 3/ 3 

ilogtan/'— +-V ilogtan 4 ^"^ - 14. ilogsec(3*+a); | log sin (4 * 2 + <* 2 ) ; 
ilogtan(* + ^. 16. -( 25 ~* 2 ) f - 

Art. 178. 3. log(*+3) 2 + — — 5. log (* 2 + 4) 2 (* - 1)' 

* — 1 

log (^±i) 3 - I tan- 1 1. 6. log^^i- 7 ' log (2 * + 5)(* - 7) 3 . 

\ * — 1 / 2 2 (* + 4) 3 

8. i*2_2* + log^+i) 2 . 9. ix 2 + log 
* — 1 




V* 2 ^! 



10. log 



+ 



(* - l) 2 

log(2*+5). 11. log &d£l*±2l . 12. log(*-3) 2 (*+3) 3 (*-2)(*+2)5 

* 



13. log(*-l) 



*-l 



14. log V4* + 5 + 



4 (4 * + 5) 



15. log * + 



ANSWERS. 479 



§log(2x + 5)+-- 16. log(x + 4) 3 V3x + 2 + ' • 17. log(x + l) 2 

X o (oX + Z) 

4s + 3 % lg j _ ^g tan -i_«_. 19 2 i g ( 3 x _ 2 ) _ i log (x 2 + 5) 
(x+1) 2 V3 

Ltan- 1 — ■ 20. log x + 2 tan"i x. 21. a? + 1 log^4J-V3 tsar 1 — . 

VE V5 * 2 V3 

22. log x 2 + V3 tan" 1 — 23. log x 3 (x 2 + 3) 2 . 24. 2 log a; - - - 2 tan" 1 -. 

V3 x 2 

25. log — ^—^ + i tan" 1 *-—t. 26. tan" 1 a; + log Vx 2 -f 1 § — 

° x 2_ 2 x + 5 2 2 & x 2 + 1 

Art. 179. 4. e x cos y ; x 3 + 4 x 2 ?/ -f 4 x — 6 ?/. 5. cos x tan y — sin x ; 

xe^ — 2 x?/ + x 2 ; 3 x — 2 x 2 - £?/ — ^-« 

2 

Page 311. 1. -i2_* + c, 1 x 2 («+ 6 > + c, r+g g «+t+3 + c> 

V2 + m + 1 w + £ + 3 

J_ rs + Cj _i2^ + 291og|, ^ 2 +8v-flog(v 2 + 3)-llV3tan-i-^ + c , 
r i s l 2 V3 

^- 2x + f log (x 2 - 2) - _L- log ^l^+ c, JL tan-i -%+ c , 

2 2V2 x + V2 6V5 2V5 

_J_ log *- 2a/ ^ + c, 7 a^ + H- <$ + *U, i sin-i — + c, 4 log (x 3 + 
4V3 + 2V3 3 

3 . ,,_,„„ ^ . . „ 1 



Vx 6 -9)+c, iz+ ° +£ log (2 3-1) +c. 2. -log sec (wix 

+ n) + c, i tan 3 x + f log (sec 3 x + tan 3 x) + 4 x + c, 00, 2.4288. 
3. x cos -1 x — Vl — x 2 + c, x sec -1 x — log (x + Vx 2 — 1) + c, x cot -1 x 

x 

+ a log (1 + x 2 ) + c, se{(log x) 2 - 2 logx + 2} + c, - ae"«(x 2 + 2 ax + 2 a 2 ) + c, 

-(x 3 + 3x 2 + 6x + 6)e-* + c, cosx(l -logcosx) + c, ^ m+ (logx-- — — ) 

wi + 1 V x m + l/ 

+ e. 4. fx^-fVx + c, 18 (^ x^ + -| xs + 1 x% + x^) + 9 log x ^ ~~ l + c, 

4 (3? - 2*) + 4 log ^-— ^, Vx 2 "^! + log (x + VW^l) + c. 5. .206 (the 

2* - 1 

base being 10), \ (l - IV 1 (e 3 - 1), - $& tt 3 . 6. - - log (m + n cos 0) + c, 

\ e 3 / w 

log (sin* tan *Uc, J- log tan (* + ^ + c, ^ ^log^^^ 

~V 2/ V2 \2 8/ 8(sec 2 x-4) 32 secx + 2 

+ c (see result in Ex. 3, Art. 118), sin"i f^J\ + c, — log 3 (wis + n) + c, 

\ 2 / 3 m 

— 1— sec-i^+c, tan-i^ + c, | log eX = c " + c, 1 log 1+^^^ + c, 
?n log a m e x + e -1 1 — tan 2 

4 V2 sin -1 ( V2 sin - j + c, cos x cos ?/ — y 2 + x + c, cos x siny + x — y + c. 



jr. 



480 DIFFERENTIAL CALCULUS. 

CHAPTER XX. 

Art. 181. 5. (6) 76. 6. 18. 8. 5. 11. - 2 ^V6. 13. (a) 2; (d) 4. 

16. .862025; 6.644025; 1.8564; .401. 17. (1) ¥ V; (2)10f; (3) 3.2; 

( 4 ) 68 T %; (5) i« 2 ; (6) 12 V2; (7) No area is bounded ; (8) (a) log 7, i.e. 

1.946 ; log 15, i.e. 2.708 ; log n ; k 2 log-. 18. ^ V|. 

Art. 182. 9. -Wtt- 10. - 3 -<V 24 7r. 11. if&Tr. 12. (a) f(2v'i - 1)tt ; 
(6) f(4^/2-l>. 13. Wtt. 18. 405 (|-|V» 226 (|-|) 

Art. 183. 2. y 2 = 48 X - 80 ; 24. 3. a - 4 = 2 log ?/. 4. x - 4 = 4 log y ; 

4. 5. 3 y 2 = 16 x. 6. 5?/ 2 = 48x 2 — 112; the conies y 2 = kx 2 + c, & and c 
denoting arbitrary constants. 7. 3 y = x 2 + 6 ; the parabolas y = kx 2 + c, 

k and c being arbitrary constants. 8. y 2 = 7 x + 4 ; the parabolas 

?/ 2 = #x + c, A; and c being any constants. 9. The circles r = c sin d ; 

r = 4 sin 0. 10. r 2 = ce# ; r 2 = 4 e#. 11. r = a(l — cos 0), in which a is an 
arbitrary constant. 

CHAPTER XXI. 

Art. 186. 1. f v^(Va— 3)+4tan-iv^+c- 2. 2(Vx-tan- 1 Vx)+c. 

3. |(3 x- 2y - +c. 4. ^(2+x) 2 (5x + 17)+c. 

3 V3 x - 2 



5. — f log(7 + 5V2 -x) + c. 6. x + 1 + 4Vx + 1 + 4 1og(Vx + 1 - l)+c. 

Art. 187. 5. | V4x 2 + 6x + 11 + £log (2 x + | + V4x 2 + 6x + 11) + c. 

6. - 3Vl2-4x-x 2 - 10 B in-i^-i^ + c. 7. -J—log^^^^5±^ + c. 

4 2 V3 V6-3x+V6 + x 



8. 3Bm-igL±J-Ai og vfr-3»-^ + g + c . 9 . log »-l + V^+»+l + c . 

4 V3 V6-3x+V6 + x x + l + Vx 2 + x+l 



x 



1 +a/x 2 + x+1 



10. Vx 2 + x + 1 + -f log(x + i + Vx 2 + x + 1) - 3 log- — x-r v^ -t^t-i + e 

aj + 2 , z+i + Vx 2 +x+l 

11. isec -1 — (- c. 

Art. 190. 2 1 cos 3 x — cos x + c ; sin x — | sin 3 x + c ; f cos 3 x — | cos 5 x 

2 8 

— cosx + c. 4. (1) f cos 3 x(cos 2 x — 4)+c; (2) 5siir 5 'x(| — isin 2 x+2V sin4x )+ c 5 

(3) 2 Vsinx (1 — | sin 2 x + | sin 4 x) + c ; (4) 3 cos 3 x( T 1 r cos 2 x — |) + c. 

7. (1) ^tan 3 x-j-tanx + c; (2)— icot 3 x — cotx + c; (3) 1 tan 5 x+|tan 3 x+tanx + c. 
9. (1) T V tan 3 x(3 tan 2 x + 5) + c ; (2)2 tan 2 xQ + f tan 2 x + ^ tan* x) + c ; 
(3) | tan 3 x(i + } tan 2 x) + c ; (4) sec 3 x( } sec 4 x - f sec 2 x + $) + c : 

(5) | Vcsc x(5 — esc 2 x) 4- c ; (6) — esc 3 x{\ esc 4 x — § esc 2 x + |) + c . 



sin 8 x\ 

"T" c - 



ANSWERS. 481 

Art. 191. 3. (1) i^-sin2z + ^_±^\ + c ; (2) ^x + 4 sin 2 x 

-isin32^ + fsin4^)+ 2 c; (3) JL_^i*_^i2x 

3 ¥ y ' w 16 64 ' 48 ' 

(4) ¥ x g cos 2 x(cos 2 2 x — 3) + c ; (5) T \j ( 3 x — sin 4 a; + 

Art. 192. 1. (1) - sin ^ cos - + - +. c ; (2) - i sin 2 a; cos x - f cos k + c ; 

(3) - cosa;smx (sin 2 x+t) + tx+c; (4) _i s in*xcosx- i^^(sin 2 x+2)+c. 
4 15 

2.(1) -cjtx + c;(2)ilogtan | - | cot x esc x + c; (3) -|-^|^ -fcotx + c. 

5. (1) |sinxcosx(2cos 2 x+3)+f£ + c; (2) | sin x(cos 4 x + f cos 2 x+|) + c ; 
(3) i smx -ff tanx+c; (4) £tanxsec 3 x+§ sec x tan x + flog (secx-f tanx)+c. 

C0S 3 X / \ 

6. (1) h tanxsecx + J log tan (7 + -) + c; (2) ± tanx (sec 2 x -f 2) + c ; 

, \4 2/ / ^ . 

(3) I tan x sec 3 x + f < tan x sec x + log tan ( - + - ) I 4- c. 7. (1) | log tan - 

— i cot x cosec x + c ; (2) — 1 cot x (cosec 2 x + 2) + c ; (3) — \ cot x cosec 3 x 

cot x cosec x — log tan - J + c. 11. (1) | tan 2 x — log secx + c ; 

(2) — icot 3 x + cotx+«+c; (3) itan 3 x— tan x+x+c; (4) itan 4 x— | tan 2 x 
+logsecx + c. 14. (1) i(sinxcosx+x)— ^sinxcos 3 x + c; (2) — | sinxcos 5 x 

-f 2V sin x cos 3 x+ -^s sin x cos x + T l g x + c ; (3) ^^ (3 — cos 2 x) + c. 

_ sin x .J 

17. (1) -icot 7 x-^cot 5 x + c; (2) Ltan 4 x+c; (3) - T Vcot 3 x(3cot 2 x+5)+c. 
^ C^H!_V3tan-i/^Ul 



Page 343. 3. (1) 3 fir + J log ^ ^- - V3 tan-* - * ^ - +c; 

* - 1 _V V3 / 

(2) Sf^+Si +c . (8) _L t . 1I . 1 /_^5 = .U e . (4 )_Ll 0g 4=M^ + c; 

8^27+1 V5 VVl-4x 2 / 2a/5 Vl-4x 2 +V5 

(5) _ 2V4x-x 2 _ vers - 1 x +c , (6)2V , a> 2 +3x+5-21og(x+f+V / x 2 +3x + 5)+c; 

(7)21og(x + f + V x 2 ^Bx + 5 ) + -^log 10 + 3 ^- 2V5 ^+ ^±^ + c; 

V5 * 

m _ 1 l0 „ l-» + V2(tt 2 +l) ■ c . rq x _ 1 f__^ 3x 2 

C) V2 * + l +C ' (9) T28 l(x*-16) 2 32(x*-16) 

3 i™-z' 2 - 4 l 1 . nm z(3x 2 + 20) 3 f ,_!« ,' . 

— -54^ log v + c ; (10) — + 2tit tan x — he 

^ 6 b a;2 + 4;^ ' ^ .' 128(x 2 + 4) 2 25b 2 ' 



CHAPTER XXII. 

Art. 195. 2. 2525. 3. 3690 ; 3660 ; (true value = 3660). 6. 333 in 
20.000. 7. .05075 ; 1509. 



482 DIFFERENTIAL CALCULUS. 



CHAPTER XXIV. 

Art. 201. 4. The parabolas y — 3 x 2 + c& + c 2 , whose axes are parallel 
to the */-axis ; 2 y = 6 x 2 + 11 * — 13 ; y = 3 se a + 15 a; + 22. 5. The cubical 
parabolas y = ce 3 -f C\X + c 2 ; y = x % 4- x ; y = x 3 — x + 4. 6. The cubical 
parabolas ?/ = ex 3 + Ci# + c 2 , in which c, Ci, c 2 are arbitrary constants ; 
6 y = x s + 11 x ; 5 y + x 1 + 16 = 22 x. 7. The cubical parabolas x = Ci?/ 3 

+ c 2 ?/ + c 3 ; 120 sb= 11 ?/ 3 - 251 «/ + 240; 7 x + 4 y* = 62 y-85. 8. 15,528 ft.; 
62.1 sec. 10. Half a mile. 

Art 202. 4. (1) 37; (2) SS^a 3 ; (3) 6 a 3 ; (4) -fa 8 *-. (5) ^a&c; 
(6) fTra 3 ; (7) ^; (8) "£; (9) * Tra 3 - | a 3 . 

Art. 203. 3. 5. 

Art. 204. 5. 1154.7 cu. in. 6. fa 3 tana. 7. |(* - f)a 3 . 8. 5440.6 
cu. in. ; 7ra 3 tan #. 

Art. 205. 4. f7r(a 2 -& 2 )£' 



CHAPTER XXV. 



Art. 


207. 4. 


301.6 


i 


irabh. 






5. 55f cu. ft. 


6. f ab 2 cot a. 


7. f (3 


7T + 8) a 3 




8. 


ia%. 










Art. 


208. 2 


7ra 2 
12 ' 


3. 


a 2 
2' 


n 


5. 


fxa 2 . 6. 11 7T. 7. 


f« 2 . 


Art. 


209. 2. 


(1) 2 


tra 


? 






(2) (5) {V2 + log(V2 + l)}a; 


(3) 4a(cos^- 


cos^ 


8 


a; (4) 


i<« 


?1 

a _ 


-"•>• !(-;>•■ 


4(a 2 + a5 + & 2 ) 
a + b 



Art. 210. 2. (3) — ; (4) (a) Z sec a, in which Z is the difference in 
A 

length of the radii vectores to the extremities of the arc ; (4) (5) like (4) (a); 



(5) % U Vl + *2 2 - h Vl + 0i* + log ^LVi±_%] ; (6) a tan £ sec | + 
/ L c/i + VI + 01- 1 z z 

a log tan (- 1 + -\ ; 2 a [sec - + log tan %^j ■ 



sin -1 e 



e 



Art. 211. 5. 4 7raV 6. ir {j - 2)a 2 . 7. 2 7r& 2 + 2 7ra5 

8. (1) 3 7ra 2 ; (2) 5 7r 2 a 3 , *£ Tra 2 ; (3) 7r 2 a 3 , A^a 2 . 9. 2ttW, 4ir 2 ab. 

10. 2 7ra 2 fl--V 12. 2 1 4 7r « 3 (3*-2); — 7ra 2 (7r + 4). 
V ej 2V2 

Art. 212. 2. 4 a 2 . 3. 4 Tra 2 . 4. Surface = 8 a ( 2 6 sin"* 



ft2 ■ Va 2 - & 2 

— a sin -1 



ANSWERS. 483 

Art. 213. 3. 1341 ; 9J. 4. 4.62. 5. (1) 2|, 5| ; (2) f, 1.14, .94 ; 

(3) 5^,9^. 6. (1) 9.425; (2) 15.71; (3) 1.5716,1.571a. 7. ^, ^. 

IT TT 

9. Arc 2 . 10. 1.273 a. 12. 1.132 a, 1.5 a 2 . 13. f a, \ a 2 . 14. 32.704°. 
15. ia, fa. 16. f a, § a 2 . 17. f a, | a 2 . 18. 1.273 a, 2 a 2 . 19. M66a,la 2 . 

CHAPTER XXVII. 



Art. 219. 2. y Vl - x 2 + x Vl - y 2 = c. 3. (?/ + &) n (x + a)" 1 = c. 

Art. 220. 1. x 2 + y 2 = cy. 2. x 2 (x 2 + 2 y 2 ) = c 4 . 3. a# 2 = c 2 (x + 2 ?/). 
4. x?/(x - y) =c. 

Art. 221. 1. x?y = c. 2. x 2 y + 3x + 2y* = c. 3. e* sin y + x 2 = c. 

4. 3 ax?/ - yi = x 3 + c. 7. a log (x 2 ?/) - y = c. 8. log — = — . 

?/ xy 

Art. 222. 3. Vl — x 2 • y = sin -1 x + c. 4. w = tan x — 1 + ce _tanx . 

1 i i 

,5. y = x-(l + ce x ). 7. 3 y* = c(l - x 2 ) 4 - 1 + x 2 . 8. y 2 (x 2 + 1 + ce* 2 ) = 1. 

Art. 223. 2. y 2 = 2 ex + c 2 . 3. 2/ = c - [p 2 + 2p + 2 log Q> - 1)], 

x = c — [2_p + 2 log (p — 1)]. 4. log (p — x) = — (- c, with the given 

p — x 

relation. 5. (x 2 + ?/) 2 (x 2 - 2 y) + 2 x(x 2 - 3 y)c = c 2 . 6. y = cx + -. 

7. ?/ = ex + a Vl + c 2 . 8. ?/ 2 = ex 2 + c 2 . c 

Art. 224. 2. x 2 + ?/ 2 = « 2 ; x 2 (x* - 4 y 2 ) = 0. 3. (1) y = cx+c 2 , 

x 2 + 4y = 0. (2) (?/ + x-c) 2 = 4x?/, x?/=0. (3) (x - y+ c) 3 = a(x + y) 2 , 
x + ?/ = 0. 

Art. 225. 3. The concentric circles x 2 + y 2 = a 2 . 4. The lines y = mx. 

8. (1) The ellipses y 2 + 2x 2 = c 2 ; (2) the hyperbolas x 2 - y 2 = c 2 ; (3) the 

4. 4 4 

conies x 2 + rc?/ 2 = c ; (4) the curves y 3 — x 3 = c 3 ; (5) the ellipses x 2 + 
2 y 2 = c 2 ; (6) the cardioids r = c(l + cos 6) ; (7) the curves r n cos w0 = c n ; 
(8) the curves r n = c n sin nd ; (9) the lemniscates r 2 = c 2 sin 2 0, whose axes 
are inclined at an angle 45° to the axes of the given system ; (10) the con- 
focal and coaxial parabolas r(l — cos 6) = 2 c ; (11) the circles x 2 + y 2 -2 Ix 
+ a 2 = 0, in which I is the parameter. 10. The conies that have the fixed 
points for foci. 11. The conies that have the fixed points for foci. 12. The 
conies & 2 x 2 ± a 2 y 2 = a 2 b 2 . 13. The hyperbola 4 xy = a 2 . 14. The parabola 
(x-y) 2 -2a(x + y) + a 2 = 0. 

Art. 226. 3. (1) y=e 2x (a cos 3 x+6 sin 3 x). (2) */ = c 1 e 2 *+c 2 e x +c 3 e 33! . 
(3) y = Cl e 4x + e 2 *(c 2 + c 3 x). (4) y = e 2x (ci + c 2 x) + e^Ccs cos 5x+C4sin 5x). 
7. (1) y = x(a cos log x + 6 sin log x). (2) ?/ = x(ci + c 2 log x). 

(3) y = x 2 (ci + c 2 log x). (4) y = Cix -1 + x(c 2 cos log x + c 3 sin log x). 

9. y = (5 + 2 x) 2 {ci(5 + 2 x)^ 2 + c 2 (5 + 2 x)"^ 2 }. 



484 DIFFERENTIAL CALCULUS. 

Art. 227. 4. (1) y = c x e ax -f c 2 e~ ax . (2) e 2cx + 2 ccie«-» = Ci 2 . 

(3) < = ^- { - (vers- 1 — - tt^ - Vase - x 2 \ . 5. The circle of radius a. 
6. (1) 2/=Ci£+(ci 2 +l)log(z-Ci.)+C2. (2) ?/ = Cilogx + c 2 . (3) 2Q/-6) 

= g x ' a + e -(x-a) # (4) y = Ci\0g(l+X)+%X-lX*+C2. 8. (1) ?/ 2 = X 2 -|-CiX+ C 2 . 

(2) \ogy=Cie x +c 2 e- x . (3) (x-Ci) 2 =c 2 Q/ 2 +c 2 ). (4) ?/ = log cos(ci-x) + c 2 . 
Page 411. (1) r=asind. (2) xe» = c(l +x+y). (3) c(2?/ 2 + 2x?/-x 2 ) 2v/ 3 

- (V3+l)x + 2y > ^ x<2 = 2 cy + c\ (5) ?/ sec a: = log (sec x + tan a) + c. 
(l-V3)x + 2*/ 

(6) 3 y = x 2 (l + x 2 )^ + ex 2 . (7) 3 x 2 + 4 xy + 5 y 2 + 5 x + 2/ = c 

(8) (x -2 c)?/ 2 = c 2 x. (9) ?/(x 2 + l) 2 = tan" 1 x + c. (10) 60y*(x + l) 2 = 

10 x 6 + 24 x 5 + 15 x 4 + c. (11) x = — P (c + a sin -1 ??), y = — ap + 

Vl-jO 2 

— (c + asin" 1 ^). (12) x + c = a log (p + Vl + p' 2 ), y = aVl+p 2 

(13) ?/ 2 = CX 2 - -^— (14) X = CX?/+C 2 . (15) ?/=r|(j9 2 +i9)+| log (2p— 1). 

c+1 
(16) */(l±cosx)=c. (17) y 2 +(x+c) 2 = a 2 ; y 2 = a 2 . (18) ?/ = cx+ Vb 2 +a 2 c 2 ; 
6 2 x 2 + a 2 ?/ 2 = a 2 b 2 . (19) 9(y + c) 2 = 4 x(x - 3 a) 2 ; x = 0. (20) y = c x e ax 
-\-c 2 e- ax + c 3 sin (ax + a). (21) ?/ = (cie z + c 2 e~ x ) cosx + (c 3 e a: + c 4 e-*) sin x. 

(22) ?/ = e 2x (ci + c 2 x) + c 3 e-*. - (23) y = dx + c 2 x~\ (24) ?/ = - 1 + 

- x 

x?jc 2 cosf — -logx) + c 3 sin( — -logxj j- (25) y = c x (x+ a) 2 + c 2 (x + «) 3 . 

(26) (cix + c 2 ) 2 + a = d?/ 2 . (27) 3 x = 2 ah (y% - 2 ci) (yi + Ci)i + c 2 . 

(28) y = ci log x + i x 2 + c 2 . (29) e-°* = c x x + c 2 . 



INDEX. 



[The numbers refer to pages.] 



Abdank-Abakanowicz, 290. 

Absolute, constants, 16 ; value, 14. 

Acceleration, 105, 223-229. 

Adiabatic curves, 86. 

Aldis, Solid Geometry, 212. 

Algebra, Chrystal's, 62, 65, 181, etc.; 
Hall and Knight's, 65, 233. 

Algebraic equations, theorems, 94, 168. 

Algebraic functions, 17, 56, 93. 

Allen, see 'Analytic Geometry.' 

Anisler's planimeter, 348. 

Analytic Geometry, Ash ton, 129 ; Candy, 
5; Tanner and Allen, 129; Went- 
worth, 129. 

Analytical Society, 39. 

Angles at which curves intersect, 81. 

Anti-derivatives, 45, 48. 

Anti-differentials, 45, 291, 292. 

Anti-differentiation, 269, 291. 

Anti-trigonometric functions, 17. 

Applications : elimination, 111 ; equa- 
tions, 93, 94, 171 ; geometrical, 79; 
physical, 79; rates, 90; of inte- 
gration, 313, etc. ; of successive 
integration , 360, etc. ; of integra- 
tion in series, 350 ; of differentia- 
tion in series, 240; of Taylor's 
theorem, 244-248, 254-256; to mo- 
tion, 214. 

Approximate integration, 344 ; by means 
of series, 353. 

Approximations : to areas and integrals, 
278, 344, 353 ; to roots of equations, 
171 ; to values of functions, 44 ; to 
small errors and corrections, 92, 
138. 

Arbitrary constants, 16. 

Arbogaste, 36. 

Arc: derivative, 98, 99; length, 370, 375, 
127; Huygheus' approximation, 
249. 



Archimedes, see ' Spiral.' 

Area, 10; approximation to, 314, 346, 
derivative, differential, 95, 97 ; me- 
chanical measurement, 318, 349; 
of curves, 313, 367, 369; of a 
closed curve, 319, 370 ; of surfaces 
of revolution, 374; of other sur- 
faces, 378 ; precautions in finding, 
319 ; sign of, 318, 370 ; swept over 
by a moving line, 370. 

Argument, 142. 

Ashton, see 'Analytic Geometry.' 

Astroid, see ' Examples.' 

Asymptotes, 199, 212, 213 ; circular, 205 ; 
curvilinear, 204; oblique, 203; par- 
allel to axes, 201 ; polar, 205 ; vari- 
ous methods of finding, 204. 

Asymptotic circle, 205. 

Bernoulli, 271. 

Binomial Theorem, 245. 

Bitterli, 290. 

Borel, divergent series, 235. 

Burraann, 19. 

Byerly, see ' Calculus.' 

Cajori, History of Mathematics, 36, 40, 
270, 325, 343. 

Calculation of small corrections, 92. 

Calculus, 1; differential, 11, 33, 270; 
integral, 11, 33, 45, 270; invention, 
1, 270; notions of, 11. 
references to works on: Byerly, 
Problems, 108, etc.; Campbell, 
225; Echols, 35, etc.; Edwards, 
Integral, 334, etc.; Edwards, 
Treatise, 127, etc.; Gibson, 41, 
etc.; Harnack, 170, etc.; Lamb, 
41, etc. ; McMahon and Snyder, 
Biff., 41, etc.; Murray, Integral, 
284, etc. ; Osgood, 170,* etc ; Perry, 



485 



486 



INDEX. 



12, 431, etc.; Smith, W. B., 133, 
343; Snyder and Hutchinson, 277, 
etc. ; Taylor, 127, etc. ; Todhunter, 
Biff., 65, etc. ; Integral, 284; Wil- 
liamson, Diff., 65, etc.; Integral, 
284, etc. ; Young and Linebarger, 
431. 

Campbell, see ' Calculus.' 

Candy, see ' Analytic Geometry.' 

Cardioid, see ' Examples.' 

Catenary, see ' Examples.' 

Cauchy, 234 ; form of remainder, 250. 

Centre of curvature, 157, 158; of mass, 
385. 

Change of variable, in differentiation, 
143; in integration, 296. 

Changes in variable and function, 30, 31. 

Chrystal, see 'Algebra.' 

Circle, curvature of, 155; of curvature, 
156 ; osculating, 152, 159 ; see ' Ex- 
amples.' 

Circular asymptotes, 205. 

functions and exponential functions, 
250. 

Cissoid, see ' Examples.' 

Clairaut's equation, 399. 

Commutative property of derivatives, 
131. 

Comparison test for convergence, 237. 

Complete differential, 134. 

Compound interest law, 65. 

Computation of it, 351. 

Concavity, 148. 

Condition for total differential, 138. 

Conjugate points, 208. 

Conoids, 366. 

Constant: absolute, 16 ;. arbitrary, 16; 
elimination of, 111; of integra- 
tion, 281, 283, 287, 395. 

Contact: of curves, 149; order of, 149; 
of circle, 150; of straight line, 
151. 

Continuity, continuous function, see 
' Function.' 

Convergence: 234, 237; interval of, 237; 
tests for, 237, 238; see 'Series,' 
■ Infinite Series.' 

Convexity, 148. 

Corrections, 92. 

Cos x, derivative of, 69; expansion for, 
245, 248. 

Criterion of integrability, 309. 

Critical point, critical value, 114, 116. 

Crossing of curves, 81, 151, 255. 



Cubical parabola, see ' Examples.' 

Curvature : 153 ; average, 154 ; at a point, 
154, 155 ; total, 154, centre of, 157, 
158 ; of a circle, 155 ; circle of, 156 ; 
radius of, 156, 159. 

Curves: area of, 313, 367, 369; asymp- 
totes, 199, 212, 213; contact of, 
149; derived, 38; differential, 38; 
envelope, 190; equations derived, 
324; evolute, 160; family, 190; 
integral, 289, 290; involutes, 164; 
length, 370, 373, 427 ; locus of ul- 
timate intersections, 191 ; Loria's 
Special Plane, 212; of one pa- 
rameter, 257, 260; parallel, 164; 
twisted, skew, 258; see 'Exam- 
ples.' 

Curve tracing, 211. 

Curvilinear asymptotes, 204. 

Cusps, 193, 206, 207, 209, 210. 

Cycloid, see ' Examples.' 

Decreasing functions, 113. 

Definite integral, see ' Integral.' 

De Moivre's theorem, 251. 

Density, 385. 

Derivation of equation of curves, 324. 

Derivative: definition, 32; notation, 35; 
general meaning, 40; geometric 
meaning, 37; physical meaning, 
39; progressive, regressive, 167; 
right and left hand, 167. 

Derivatives : of sum, product, quotient, 
46, 48-52; of a constant, 47; of 
elementary functions, 56-75 ; of a 
function of a function, 54; of im- 
plicit functions, 75, 137; of in- 
verse functions, 56; special case, 
55; geometric, 95-102; successive, 

103, 108 ; meanings of second, 

104, 105. 

Derivatives, partial, 76, 128, 129; com- 
mutative property of, 131; geo- 
metrical representation, 130; il- 
lustrations, 139-142; successive, 
131. 

Derivatives, total, 134 ; successive, 139. 

Derived, curves, 38; functions, 32, 34. 

Descartes, 270. 

Difference-quotient, 32, 34. 

Differentiable, 35. 

Differential calculus, see ' Calculus.' 

Differential coefficient, see 'Derivative.' 

Differential, differentials, 42, 44; com- 



INDEX. 



487 



pie te, 134; exact, 138; geometric, 
95-102; infinitesimal, 276; par- 
tial, 134; successive, 109; total, 
134, 135; illustrations, 139-142; 
condition for total, 138; integra- 
tion of total, 309. 

Differential equations, 112, 394; classifi- 
cation, 394 ; Clairaut's, 399; exact, 
390; homogeneous, 396, linear, 
397, 406, 408; order, 394; ordi- 
nary, 394; partial, 394; second 
order, 409; solutions, 112, 395, 
400 ; references to text-books, 112, 
411, etc. 

Differentiation, 33, 291 ; general results, 
46; logarithmic, 63 ; of series, 240; 
successive, 103; see 'Derivative,' 
' Derivatives.' 

Direction cosines of a line, 258. 

Discontinuity, discontinuous functions, 
see 'Functions.' 

Displacement, 214, 216, 218. 

Divergent series, see ' Series.' 

Double points, 193, 206, 207. 

Doubly periodic functions, 342. 

Durand's rule, 348. 

Echols, see ' Calculus." 

Edwards, see 'Calculus.' 

Elementary integrals, 293, 301. 

Elimination of constants, 111. 

Ellipse, see 'Examples.' 

Ellipsoid, 360. 

Elliptic functions, 279, 342. 
integrals, 279, 342, 354. 

End-values, 276. 

Envelopes, contact property, 193, 195; 
definition, 191; derivation, 194, 
197. 

Equations, approximate solution of, 171 ; 
derivation of, 324; graphical rep- 
resentation, 19, 128; roots of, 94, 
171 ; of tangent and normal, 83. 

Equiangular spiral, see ' Examples.' 

Errors, small, 92, 136; relative, 92, 136. 

Euler, 139, 251, 351; theorem on homo- 
geneous functions, 139. 

Evolute, definitions, 160. 
properties of, 161. 

Evolute of the ellipse, see 'Examples.' 

Exact differential, 138. 
equations, 396. 

Examples concerning : 
adiabatic curves, 86. 



astroid (or hypocycloid) , 85, 98, 158, 

161, 319, 324, 376, 405, 425. 
cardioid, 90, 97, 159, 369, 374, 377, 389, 

405, 425, 433, 446. 
catenary, 322, 373, 378, 426, 433. 
circle, 85, 159, 315, 369, 374, 377, 388, 

389, 391, 404, 449. 
cissoid, 203. 
cubical parabola, 91, 97, 98, 158, 279, 

287, 316, 319, 322. 
cycloid, 86, 158, 161, 373, 377, 426. 
ellipse, 85, 102, 164,203, 321, 324, 373, 

382, 387, 435, 447, 449, 450. 
evolute of the ellipse, 161, 164. 
exponential curve, 85. 
folium of Descartes, 86, 203, 369. 
harmonic curve, 448. 
helix, 328, 329. 
hyperbola, 86, 91, 158, 159, 161, 203, 

204, 212, 405, 433. 
hypocycloid, see ' Astroid.' 
lemniscate, 159, 369, 405, 433. 
limacon, 448. 
parabola, 85, 86, 91, 98, 100, 158, 159, 

161, 164, 196, 197, 203, 213, 273, 

280, 287, 316, 317, 319, 359, 374, 

382, 389, 405, 426, 433. 
probability curve, 203. 
semi-cubical parabola, 85, 86, 158, 

280, 319, 426. 
sinusoid, 85, 280. 
tractrix, 426. 
the witch, 86, 159, 203. 
Spirals : 

Archimedes', 90, 97, 99, 159, 374. 

equiangular (or logarithmic), 90, 
159, 369, 374, 426. 

general, 90, 159. 

hyperbolic (or reciprocal), 90, 
369. 

logarithmic, see ' Equiangular.' 

parabolic (or lituus) , 90. 

reciprocal, see 'Hyperbolic' 
Expansion of : 

cos x, 245, 248. ■ 

log (1 + x), logarithmic series, 244, 

352. 
sin x, 245, 248. 
sin- 1 x, 351. 

e' c , exponential series, 249. 
tan- 1 ^, Gregory's series, 350. 
Expansion of functions : 
by algebraic methods, 249. 
by differentiation, 240. 



488 



INDEX. 



Expansion of functions : 

by integration, 350. 

by Maclaurin's series, 247-249. 

by Taylor's series, 243-247. 
Explicit function, 16. 
Exponential curve, see 'Examples.' 

function, 17; expansion of, 249 ; and 
trigonometric, relations between, 
250. 
Extended Theorem of Mean Value, 177. 

Family of curves, 190. 

Fermat, 120, 270, 372. 

Fluent, fluxion, 39. 

Folium of Descartes, see ' Examples.' 

Forms, indeterminate, 180. 

Formulas of reduction, 334, 339. 

Fourier, 276. 

Fractions, rational, integration of, 305. 

Frost, Curve Tracing, 204, 206, 212. 

Function, 14; algebraic, 17,56, 342; cir- 
cular, 342; classification, 16; con- 
tinuous, 18, 25, 35, 129; derived, 
32, 34; discontinuous, 18, 25, 27; 
elliptic, 279, 342; explicit, 16; 
exponential, 17, 01 ; graphical 
representation, 19, 20, 128; homo- 
geneous, Euler's theorem on, 139; 
hyperbolic, 304, 342, 413 ; implicit, 
10, 75, 137 ; increasing and decreas- 
ing, 113; inverse, 15, 56, 71; irra- 
tional, 17, 327 ; logarithmic, 17, 61 ; 
many-valued, 15; march of a, 
121 ; maximum and minimum val- 
ues of, 114; notation for, 18; of a 
function, 54, 55; of two variables, 
16, 128; one-valued, 15; periodic, 
342; rational, 17, transcendental, 
17; trigonometric and anti-trigo- 
nometric, 17,66, 336; turning val- 
ues of, 115 ; variation of, 115. 

Gauss, 234. 

General integral, see ' Integral.' 

spiral, swe 'Examples.' 
Generalized Theorem of Mean Value, 

182. 
Geometrical interpretation, a certain, 

336. 
Geometrical representation of: 

derivatives, ordinary, 37. 

derivatives, partial, 130. 

functions of one variable, 19. 

functions of two variables, 128. 



function of a function, 55. 

integrals, definite, 284. 

integrals, indefinite, 287. 

total differential, 135. 
Geometric derivatives and differentials, 

95-102. 
Gibson, see ' Calculus.' 
Glaisher, Elliptic Functions, 343. 
Goursat-Hedrick, Mathematical Anal- 
ysis, 170. 
Graphical representation of functions, 

19 ; of real numbers, 13. 
Graphs, sketching of, 121. 
Gregory, 235, 351. 
Gregory's series, 351. 
Gyration, radius of, 390. 

Harkness and Morley, Analytic Func- 
tions, 233, Theory of Functions, 35. 
Harmonic curve, 448. 
Harmonic motion, 78, 107. 
Harmonic series, 234. 
Harnack, see ' Calculus.' 
Hele Shaw, Mechanical Integrators, 349. 
Helix, 258, 428. 

Henrici, Report on Planimeters , 349. 
Herschel, 19, 40. 

Hobson, Trigonometry, 233, 352, 423. 
Homogeneous, differential equations, 
396. 

functions, Euler's theorem, 139. 

linear equation, 397, 406, 408. 
Horner, Horner's process, 247, 256. 
Hutchinson, see ' Calculus.' 
Huyhen's rule for circular arcs, 249. 
Hyperbola, see 'Examples.' 
Hyperbolic functions, 304, 342, 413. 

.spiral, see 'Examples.' 
Hypocycloid, see 'Examples.' 

Implicit functions, 16; differentiation, 
75, 137. 

Increasing function, 113. 

Increment, notation for, 4, 30, 31. 

Indefinite integral, see ' Integral.' 

Indeterminate forms, 180. 

Inertia, centre of, 386. 
moment of, 390. 

Infinite numbers, 14, 28, 29. 
orders of, 29. 

Infinite series, 230; algebraic proper- 
ties, 234; differentiation of, 232, 
240 ; general theorems, 235 ; inte- 
gration in, 353; integration of, 



INDEX. 



489 



232, 350; limiting value of, 231; 
questions concerning, 231 ; Osgood, 
article and pamphlet, 233, 236, 237 ; 
remainder, 236 ; study of, 233. 
See ' Series.' 

Infinitesimal, 1, 28, 43, 45. 

Infinitesimal differential, 276. 

Infinitesimals, 28; orders, 29; summa- 
tion, 271. 

Inflexion, points of, 116, 125, 127. 

Inflexional tangent, 127. 

Integral curves, 289, 290. 

Integral, definite, approximation, 344, 
353; definition, representation of, 
properties, 275-279, 284, 285. 

Integral: double, 355; element of, 276: 
elementary, 293, 301 ; elliptic, 279, 
342, 354, 373; general, 283. 

Integral, indefinite, 281, 283; represen- 
tation of, 287. 

Integral : multiple, 356; particular, 283; 
precautions in finding, 319 ; triple, 
355. 
See ' Calculus.' 

Iutegraud, 271. 

Integraph, 290, 348, 349. 

Integrating factors, 396. 

Integration, 269, 291 ; as summation, 
275, 291 ; as inverse of differentia- 
tion, 281, 291; constant of, 281, 
283, 287 ; general theorems in, 294 ; 
successive, 355, 357. 

Integration: by parts, 298; by substitu- 
tion, 296, 304, 328, 336 ; by infinite 
series, 350, 353; by mechanical 
devices, 318. 

Integration of: infinite series, 232, 350; 

irrational functions, 327 ; rational 

fractions, 305; total differential, 

309; trigonometric functions, 336. 

See ' Applications.' 

Integrators, 348, 349. 

Interpolation, 256. 

Intrinsic equation, 374, 423. 

Invention of the calculus, 1, 270. 

Inverse functions, 15, 56, 71. 

Involutes, 164. 

Irrational functions, integration, 327. 

Isolated points, 206, 208. 

Jacobi, 131. 

Kepler, 120. 
Klein, 62. 



Lagrange, 36, 249, 270. 

Lagrange's form of remainder, 250. 

Lamb, see ' Calculus.' 

Laplace, 270. 

Legendre, 354. 

Leibnitz, 36, 39, 195, 270, 271, 351. 

theorem on derivative of product, 
110. 

Lemniscate, see ' Examples.' 

Lengths of curves, 370, 373, 427 ; of tan- 
gents and normals, 84, 88. 

Limacon, see 'Examples.' 

Limits, limiting value, 20, 23, 36 ; in in- 
tegration, 276; of a series, 231. 

Linear differential equations: of first 
order, 397 ; with constant coeffi- 
cients, 406; homogeneous, 408. 

Linebarger, see ' Calculus.' 

Lituus, see 'Examples.' 

Locus of ultimate intersections, 191. 

Logarithmic, differentiation, 63. 
function, 17, 62. 
series, 244, 352. 
spiral, see 'Examples.' 

Loria, Special Plane Curves, 212. 

Machin, 351. 
Maclaurin, 250. 

theorem and series, 247, 252. 
Magnitude, orders of, 29. 
Mass, centre of, 385. 
Mathews, G. B., 235. 
Maxima and minima, 113 ; by calculus, 
114-120; by other methods, 120; 
of functions of several variables, 
120 ; practical problems, 121. 
McMahon, proof, 138. 

See ' Calculus.' 
Mean values, 380. 
Mean value theorems : 

differentiation, 164, 169, 174-179, 182. 

integration, 286, 380. 
Mechanical integrators, 348. 
Mechanics, 385. 

Mellor, Hie/her Mathematics, 431. 
Mercator, 352. 
Minima, see ' Maxima.' 
Moment of inertia, 390. 
Morley, see ' Harkness.' 
Motion, applications to, 214. 
Motion, simple harmonic, 78, 107. 
Muir, on notation, 131. 
Multiple, angles in integration, 338. 

integrals, 356. 



490 



INDEX. 



Multiple, points, 206, 209. 
roots, 93. 

Neil, 371. 

Newton, 39, 171, 351. 

Nodes, 207. 

Normal, equation of, 83; length, 84, 88. 

Notation for: absolute value, 14; de- 
rivatives, 35, 39, 103; differentials, 
42; functions, 18; increment, 4; 
infinite numbers, 14; integration, 
270, 283, 284, 358; inverse func- 
tions, 19, 50 ; limits, 23 ; partial 
derivatives, 76, 129, 131, 135; sum- 
mation, 276. 

Notation, remark on, 36. 

Numbers, 13 ; finite, infinite, infini- 
tesimal, 14, 28; transcendental, 62. 
e and v, 62, 328. 
graphical representation, 13, 14. 

Oblique axes, 314. 
Order of, contact, 149. 

derivative, differential, 104, 256. 

differential equation, 394. 

infinite, 29. 

infinitesimal, 29, 256. 

magnitude, 29. 
Orthogonal trajectories, 401, 403. 
Oscillatory series, 234. 
Osculating circle, 152, 159. 
Osgood, W. F., pamphlet, 233, etc. ; see 

' Calculus.' 

Parabola, see ' Examples.' 
Parabolic rule, 346. 

spiral, see ' Examples.' 
Parallel curves, 164. 
Parameter, 190, 257. 
Partial derivative, see ' Derivative.' 
Partial fractions, 305. 
Particular integral, see ' Integral.' 
Pendulum time of oscillation, 354. 
Periodic functions, 312. 
Perry, on notation, 131. 

See ' Calculus.' 
Picard, 277. 
Pierpont, Theory of Functions, 14, 27, 

131, 167. 
Planimeters, 348, 349. 

Henrici, Report on, 349. 
Points, see 'Critical,' 'Double,' 'Iso- 
lated,' 'Multiple,' 'Salient,' ' Sin- 
gular,' ' Stop,' ' Triple,' 'Turning.' 



Power series, 237, 240, 350. 
Precautions in integration, 319. 
Probabilities, 249. 
Probability curve, see ' Examples.' 
Progressive derivative, 167. 

Radius of curvature, 156, 159. 

of gyration, 390. 
Rate of change, 11, 39, 40, 41, 90. 

variation, 132. 
Rational fraction, integration, 305. 
Reciprocal spiral, see ' Examples.' 
Rectification of curves, 371. 
Reduction formulas, 334, 339. 
Regressive derivative, 167. 
Relative error, 92. 
Remainder after n terms, 236. 
Remainders in Taylor's and Maclaurin's 

series, 243, 246, 250. 
Right- and left-hand derivatives, 167. 
Ring, 323. 
Rolle, 169. 

Rolle's theorem, 166, 168. 
Roots of equations, 94, 171. 
Rouche et Comberousse, 371. 
Rules for approximate integration, 344, 
346, 348. 

Salient points, 208. 

Schlomilch-Roche's form of remainder 
250. 

Second derivative : 

geometrical meaning, 104. 
physical meaning, 105. 

Semi-cubical parabola, 371. 
See ' Examples.' 

Series, 65 ; absolutely convergent, 235 ; 
conditionally convergent, 235 ; 
convergent, 234; divergent, 234, 
235 ; harmonic, 234 ; oscillatory, 
234. 
See ' Convergence,' ' Expansion,' ' In- 
finite Series,' 'Power Series.' 

Serret, 320. 

Skew curves, tangent line, and normal 
plane, 258-261, 266-268. 

Sign of area, 31«, 370. 

Simpson, Simpson's rule, 346. 

Sin x, sin - ice, expansions, 245, 248, 351. 

Singly periodic functions, 342. 

Singular points, 206, 208. 

Singular solution, 400. 

Sinusoid, see 'Examples.' 

Slope, 5, 6, 11, 38, 79, 87. 



INDEX. 



491 



Slopes, curve of, 38. 

Smith, C, Solid Geometry, 212, 37* 

Smith, W. B., Infinitesimal Ana 

133. 
Snyder, see 'Calculus.' 
Solution, see 'Differential Equation.' 
Speed, 2, 3, 4, 214. 
Sphere, surface, 377, 379. 

volume, 324, 362, 363. 
Spiral, see ' Examples.' 
Stationary tangent, 127. 
Stirling, 250. 
Stop points, 208. 
Subnormal, rectangular, 84. 

polar, 88. 
Substitutions in integration, 296, 304, 

328, 336. 
Subtangent, rectangular, 84. 

polar, 88. 
Successive differentiation, 103. 

derivatives, 103, 108. 

differentials, 109. 

integration, 355, 357. 

of a product, 110. 

total derivatives, 134. 
Summation, examples, 271. 

integration as, 275. 
Surfaces, applications of differential 
calculus, tangent lines, tangent 
plane, normal, 262-205. 

areas of, 374, 378. 

volumes, 320, 360, 363, 365. 

Tangent, 5 ; equation of, 83 ; inflexional, 
127; length, 84, 88; stationary, 
127 ; to twisted curve, 259, 266 ; to 
surface, 262. 
Tanner, see 'Analytic Geometry.' 
Taylor, F. G., see ' Calculus.' 
Taylor's theorem and series: 

applications: to algebra, 256; to cal- 
culation, 44, 135, 245, 246, 247; to 
contact of curves, 255 ; to maxima 
and minima, 254. 
approximations by, 44, 135, 245. 
expansions by, 244-247. 
for functions of one variable, 44, 242, 

243, 246, 252. 
for functions of several variables, 

250. 
forms of, 243, 244, 246. 



historical note, 249. 
Test-ratio, 238. 
Time-rate of change, 39, 133. 
Todhunter, see ' Calculus.' 
Total derivative, 134. 

differential, 134. 

rate of variation, 132. 
Tractrix, see ' Examples.' 
Trajectories, orthogonal, 401, 403. 
Transcendental functions, 17. 

numbers, 13. 
Trapezoidal rule, 344. 
Trigonometric functions, direct and in- 
verse, 17, 71. 

differentiation of, 66-75. 

integration of, 336. 

relations with exponential, 250. 

substitutions by, 328. 
Trigonometry, Hobson, 233, 352, 423. 

Murray, 71, etc. 
Triple points, 207. 
Turning points, values, 115. 
Twisted curves, see ' Skeio curves.' 

Undulation, points of, 126. 

Value, see 'Average,' 'Limits,' 'Maxi- 
mum,' 'Mean,' 'Turning.' 

Value of it, computation of, 351, 352. 

Van Vleck, E. B., 235. 

Variable, dependent, independent, 13, 15. 
change of, 143. 

Variation, continuous variation, inter- 
val of variation, 24. 

Variation of functions, 113. 
total rate of, 132. 

Veblen-Lennes, Infinitesimal Analysis, 
14, 27, etc. 

Velocity, 91, 214-222. 

Volumes, methods of finding, 320, 360, 
363, 365. 

Wallis, 270, 371. 

Wentworth, see 'Analytic Geometry.' 

Whittaker, Modern Analysis, 234, 

239. 
Williamson, see 'Calculus.' 
Witch of Agnesi, see ' Examples.' 
Wren, 372. 

Young, see 'Calculus.' 



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